First Order Operator Expansion

As we consider schemes involving compositions of three splitting pieces, the minimum number of stages to our methods (in order for it to be consistent) will be three. The perturbed Fokker-Planck operator for a splitting method with three stages will be computed as 

In general the terms in this expansion will lead to complicated expansions. In the case of deterministic systems, the operators are always first order due to equality of mixed partials (the 2nd order derivatives are cancelled). Due to the second derivatives present in the q operator, derivatives of all orders will appear in the 

A simple first order example of a Langevin dynamics integrator is the method obtained by composing one of the Symplectic Euler variants with an Ornstein-Uhlenbeck step. To get a feel for how the expansion goes, let us work out its terms for such a splitting scheme in the case of a one degree of freedom model with unit mass H = + f/(g). To be explicit, let us say our numerical method first solves 

It is clear that even writing out the 0 h ) terms considerably increases the complexity of the right hand side. 

The commutators are best considered in relation to their action on a function (or rather, their action on a density) p(q,p). For example, we may compute directly 

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However, as Cf is a first order operator, this series involves high order mixed partial derivatives, and the convergence of the expansion has to be considered in connection with its application to given initial data, that is, we should consider the expansion... 

The Greek indices a,j3= II, B,G) count colors, the Latin indices i = u,d,s count flavors. The expansion is presented up to the fourth order in the diquark field operators (related to the gap) assuming the second order phase transition, although at zero temperature the transition might be of the first order, cf. [17], iln is the density of the thermodynamic potential of the normal state. The order parameter squared is D = d s 2 = dn 2 + dG 2 + de 2, dR dc dB for the isoscalar phase (IS), and D = 3 g cfl 2,... 

For the numerical implementation of the QCL equation, the Liouville operator S is decomposed into a zero-order part Sq which is easy to evaluate and a nonadiabatic transition part if whose evaluation is difficult. The splitting suggests that we (a) employ a short-time expansion of the full exponential by use of a first-order Trotter formula... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

A systematic route to achieve a mixed quantum classical description of EET may start with the partial Wigner representation p(R,P t) of the total density operator referring to the CC solvent system. R and P represent the set of all involved nuclear coordinates and momenta, respectively. However, p(R,P t) remains an operator in the space of electronic CC states (here 4>o and the different first order of the -expansion one can change to electronic matrix elements. Focusing on singly excited state dynamics we have to consider pmn( It, / /,) = 4>m p R, P t) 4>n) which obeys the following equation... 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

Using either the exact form of the projection operator as a function of V and V or cutting the series expansion, allows us to construct different approximate variation schemes with matrix elements of V as variables. On the other hand, inserting the expansion for the projection operator eq. (1.107) in the Schrodinger equation for the projection operator eq. (1.95) with the perturbed Hamiltonian gives in the first order ... 

The difference between H(t) and /70(/,) stems from the kinetic energy operator in the adiabatic Hamiltonian H0, which can be treated as a perturbation. Using the Cambell-Baker-Hausdorff expansion, to the first order we have... 

Within the real-space method, the kinetic energy operator is expressed by the finite-difference scheme. Here, we derive the matrix elements for the kinetic energy operator of one dimension in the first-order finite difference. By the Taylor expansion of a wavefunction i/r (/) at the grid point Z we obtain the equations,... 

The second instance leading to exponential decay of E(t) follows from a more detailed specification of the model. The oscillator-bath coupling term V is a function of oscillator and bath coordinates and can be expanded in powers of q. Matrix elements between oscillator states n) and m) of the zeroth-order term in this expansion will vanish, and so this term does not contribute to the rate constants. The leading order term, then, is first order in q. Defining the bath operator F by F = — (3V/3q) q=o, the rate constants are approximately given by... 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory ... 

The same expression applies if F is an infinitesimal Hermitian operator equal to eG, for, as e -> 0, the series expansion for an exponential yields, to first-order in ,... 

Assuming that the SO coupling is weak compared to the electrostatic interactions, we terminate the expansion after the first order. The individual SO operators in (24) can be written as... 

According to (42), the T2 mode is not JT active (in first order) in E states. However, the matrix elements of the SO operator with non-relativistic electronic wave functions vanish for a state in symmetry. It is then essential to take account of the leading nonvanishing terms in the Taylor expansion of the matrix elements of the SO operator. As shown below, these are of first order of vibrational displacements of T2 symmetry, which implies the existence of a purely relativistic E xT JT effect [34]. Linear E xT vibronic coupling is not accounted for by the JT selection rules [2] it is, therefore, a novel type of JT effect. 

To first order in the perturbation h, AAvar must be zero, while to second order it must be positive for arbitrary h. In order to calculate AAvar and the density operator on which it depends, up to the second order, we can use the perturbation expansion... 

The RS formulas for the energy expansion are well known and are given in many places (e.g., Ref. 22). A thorough development of the wave-reaction operator perturbation theory has been presented by Low-din.23 Using conventional first quantized operators, we may write down the expressions for the nth-order energy E(n), for instance, as... 

We express contributions to ft(n) diagrammatic ally, but they can be automatically identified through Eq. (23) with contributions to the (n). These are shown for the first- and second-order MBPT wave function in Fig. 1. In the wave operator expansion, we encounter disconnected but linked diagrams, i.e., diagrams that are composed of two or more separate parts like... 

This is called the first-order variational space which is spanned by the reference state, the orthogonal complement states and the single excitation states (10>, ln>, T 10> or equivalently by the set of expansion CSFs and the single excitation states n>,r 0>. The parameters K and p may be determined by minimizing the expectation value of the Hamiltonian operator within this non-orthonormal basis. This results in the non-orthogonal matrix eigenvalue equation... 

For small wavefunction expansions where an explicit determinantal representation of the wavefunction may be constructed, it is straightforward to determine the solutions to Eq. (241) by constructing the overlap matrix of the first-order variational space. This is required in the solution of the variational super-CI equations for which these linear dependences must be explicitly identified and eliminated For larger CSF expansion lengths or large orbital basis sets, however, this step could easily dominate the entire iterative procedure, so alternate methods must be examined. The solutions to Eq. (241) may be determinedby first operating from the left with a CSF... 

Figure 8.3 Reaction operating curves for a single first-order reaction of the form A products effect of expansion factor.

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