Computational irreducibility

We have already seen at least one example of a physical system capable of acting as a universal computer in our discussion of Predkin and Toffoli s [fredkin82] 

A real number x is said to be computable if there exists a computer program V such that when a positive rational number is inputted, V outputs a rational number r with ] X — r ] and halts [gerochSG]. Thus, 2, tt, exp(l), Vtt , etc. are all computable numbers. 

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The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

In order to obtain this savings in the computational cost, orbitals are symmetry-adapted. As various positive and negative combinations of orbitals are used, there are a number of ways to break down the total wave function. These various orbital functions will obey different sets of symmetry constraints, such as having positive or negative values across a mirror plane of the molecule. These various symmetry sets are called irreducible representations. 

A complete decomposition of the ab initio computed CF matrix in irreducible tensor operators (ITOs) and in extended Stevens operators. The parameters of the multiplet-specific CF acting on the ground atomic multiplet of lanthanides, and the decomposition of the CASSCF/RASSI wave functions into functions with definite projections of the total angular momentum on the quantization axis are provided. 

From a computational viewpoint, the presence of recycle streams is one of the impediments in the sequential solution of a flowsheeting problem. Without recycle streams, the flow of information would proceed in a forward direction, and the cal-culational sequence for the modules could easily be determined from the precedence order analysis outlined earlier. With recycle streams present, large groups of modules have to be solved simultaneously, defeating the concept of a sequential solution module by module. For example, in Figure 15.8, you cannot make a material balance on the reactor without knowing the information in stream S6, but you have to carry out the computations for the cooler module first to evaluate S6, which in turn depends on the separator module, which in turn depends on the reactor module. Partitioning identifies those collections of modules that have to be solved simultaneously (termed maximal cyclical subsystems, loops, or irreducible nets). 

In tearing, the objective is to wind up with less computation time required to solve the torn system compared with the time required to solve the entire block of equations simultaneously. However, the criteria for evaluating the effectiveness of the tearing are by no means so well defined as those for partitioning, where the objective is clearly to obtain the smallest possible subsystems of irreducible equations. There is no general method for determining the time needed to effect a solution of a set of equations it is necessary to consider the particular equations involved. Any feasible method of tearing, then, must be based on criteria that are related to the solution time. Some of the more obvious criteria are ... 

Because the traces contain sufficient information to decompose Ftot into irreducible representations, it is necessary to compute only the diagonal elements of the matrices of the representation. If a particular atom changes position under a symmetry operation, its displacements can contribute no diagonal elements to the matrix therefore, for that symmetry operation, such an atom may be ignored. For example, the displacements of the hydrogen atoms in water do not contribute to the character of C2 in Ftot- The displacement of Hi means that the elements (1,1), (2,2) and (3,3) of the matrix are zero. 

It can be easily checked by direct computation that we have really obtained a realization of the Lie algebra g in a Hilbert (Fock) space, [T a, T fc] = ifabc fc, in accordance with (11), where Ta = T f/aL . For an irreducible representation R, the second-order Casimir operator C2 is proportional to the identity operator I, which, in turn, is equal to the number operator N in our Fock representation, that is, if T" —> Ta, then I /V 5/,/a . Thus we obtain an important for our further considerations constant of motion N ... 

The equilibrium structure is considered of flexible polymer chains within the RIS model. This model is solved by an irreducible tensor method which is somewhat different from, and simpler than, the approach of Flory and others. The results are used to compute the light scattering intensities from dilute solutions of flexible polymer chains, and the angle dependence is found... 

Theoretical estimates of the three-body moments may be obtained from the well-known pair dipole moments. These do not include the irreducible three-body components which are poorly known. Interestingly, in every case considered to date, the computations of the three-body spectral moments y[3 are always smaller than the measurements, a fact that suggests significant positive irreducible three-body dipole components for all systems hitherto considered [296, 299] further details may be found in Chapter 5. 

For easy reference we also plot theoretical three-body moments of H2-H2-H2 which are computed from first principles based on the assumption of a pairwise-additivity, Figs. 3.46 and 6.2 (heavy curves). The pair dipole moments have been shown to allow a close reproduction of measured binary spectra from first principles in the hydrogen fundamental band, for temperatures from 20 to 300 K these are believed to be reliable. Interestingly, the pairwise-additive assumption is not sufficient to reproduce the experimental three-body moments from theory, except perhaps at the lowest temperatures. With increasing temperature, rapidly increasing differences between measured and computed moments are observed, a fact which suggests the presence of an irreducible three-body dipole component of the overlap-induced type [296]. 

In recent molecular dynamics studies attempts were made to reproduce the shapes of the intercollisional dip from reliable pair dipole models and pair potentials [301], The shape and relative amplitude of the intercollisional dip are known to depend sensitively on the details of the intermolecular interactions, and especially on the dipole function. For a number of very dense ( 1000 amagat) rare gas mixtures spectral profiles were obtained by molecular dynamics simulation that differed significantly from the observed dips. In particular, the computed amplitudes were never of sufficient magnitude. This fact is considered compelling evidence for the presence of irreducible many-body effects, presumably mainly of the induced dipole function. 

Ternary moments have been computed for several systems of practical interest [314, 422]. Recent studies are based on accurate ab initio pair dipole surfaces obtained with highly correlated wavefunctions. Because not much is presently known about the irreducible ternary components, it is important to determine to what extent the measured three-body spectral components arise from pairwise-additive contributions [296, 299]. 

Figure 1 Comparison of observed ( , ) and computed ternary spectral moments of collision-induced absorption of compressed hydrogen in the H2 fundamental band. The dashed curve represents the moments computed on the basis of the pairwise additive dipole components only. The solid line also accounts for the irreducible dipole components from Ref. [53]. (This figure is an update of Fig. 3.46.)...

We thus have an explicit expression for an energy, which can be thought of as the energy of interaction between two states described by the wave functions Wi and y/r If the integral that occurs in the numerator of the left-hand side of this equation is in fact required to have a value identically equal to zero, it will be helpful to know this at the earliest possible stage of a calculation so that no computational effort will be wasted on it. This information may be obtained very simply from a knowledge of the irreducible representations to which the wave functions y/, and y/j belong. 

It will be obvious from the content of Chapter 5 why such combinations are desired. First, only such functions can, in themselves, constitute acceptable solutions to the wave equation or be directly combined to form acceptable solutions, as shown in Section 5.1. Second, only when the symmetry properties of wave functions are defined explicitly, in the sense of their being bases for irreducible representations, can we employ the theorems of Section 5.2 in order to determine without numerical computations which integrals or matrix elements in the problem are identically zero. 

The conceptual simplicity of the CASSCF model lies in the fact that once the inactive and active orbitals are chosen, the wave function is completely specified. In addition such a model leads to certain simplifications in the computational procedures used to obtain optimized orbitals and Cl coefficients, as was illustrated in the preceding chapters. The major technical difficulty inherent to the CASSCF method is the size of the complete Cl expansion, NCAS. It is given by the so-called Weyl formula, which gives the dimension of the irreducible representation of the unitary group U(n) associated with n active orbitals, N active electrons, and a total spin S ... 

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