Expansion of the Operator

Consider a Langevin splitting method using the A, B and O splitting pieces, where we compose exact solves of each of the vector fields in a prescribed sequence to define our discretization scheme. Because we solve each piece exactly, we are able to write down the action of each individual solve on the initial distribution in terms of the vector field s evolution operator (using the respective generators for each piece as in [99]), where 

We will find it more convenient to work with the adjoint of the perturbed generator, denoted u, which we may compute as 

Under the propagation of the numerical scheme (or the discretized dynamics), 

The terms of the perturbation expansion for D can be computed using the Baker-Campbell-Hausdorff (BCH) expansion already introduced in Chap. 3. Recall that for linear operators X and Y, we can write the composition of their exponentials as 

To offer some reassurance, let us define the evolution operator for our numerical method as 

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In order to compare our approach with other approaches dealing with adiabatic corrections we perform simple model calculations for adiabatic corrections to ground state energy. We start with adiabatic Hamiltonian (32). We now perform the following approximation. We limit ourselves to finite orders of Taylor expansion of the operators H and H g We shall use similar approximation as in [25]. The diagrammatic representation of our approximate Hamiltonian will be... 

In summary, the techniques described in this chapter allow us to derive expansions of the operators that correspond to physical quantities, in terms of irreducible tensors in the spaces of orbital, spin and quasispin momenta, and also to separate terms that can be expressed by operators whose eigenvalues have simple analytical forms. Since the operators of physical quantities also contain terms for which this separation is impossible, the following chapter will be devoted to the general technique of finding the matrix elements of quantities under consideration. 

These operators must be supplemented by intrashell operators whose expansion has been derived in Chapter 15. Comparison of the expansion of the operator in the quasispin space of two shells (17.71) with that of the same operator in the quasispin space of each shell individually (17.44) shows that many terms of expansion (17.71) have a much simpler tensorial structure and a smaller number of operators being summed up than those of (17.44). 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

Explicit expressions for the Van der Waals constants may be obtained by invoking the well-known147 multipole expansion of the operator V. In an arbitrary space-fixed coordinate system, this expansion can be written as... 

In the following considerations we will need the multipole expansion of the operator rl2 as series of products of operators depending on the coordinate of the particle 1 with respect to a center a, iq, of the particle 2 with respect to another center b, r2, and on the coordinates describing the relative position of the centers a... 

Inserting the multipole expansions of the operators rfj1 and rfA with respect to the pairs of sites (a, b) and (a1, b ), respectively, and defining the distributed polarizability tensor,... 

Several methods can be distinguished within the framework of the perturbative approach. Some [29-37] are based on a multipolar expansion of the operator i.e. the interaction potential of the two species, others rely on the linear response theory [38,39]. 

To progress further toward practical implementation, specific choices must be made for how one is going to approximate the neutral molecule wave function lO, N) and at what level one is going to truncate the expansion of the operator Q K) given in Eq. (5). It is also conventional to reduce Eq. (7) to a matrix eigenvalue equation by projecting this equation onto an appropriately chosen space of A + 1-electron functions. Let us first deal with the latter issue. 

The expansion of the operator sin(ft>iF/ ai2) /Ran on the spherical harmonics generates the decay probability multipole expansion. It can be written in the following known form ... 

In January 2011 the Boliden board decided to invest SEK 3.9 billion in the expansion of the operation in Garpenberg, with the aim of increasing the production to 2.5 Mt/yr. 

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