Reducing a representation

It was discussed before that the irreducible representations can be produced from the reducible representations by suitable similarity transformations. Another important point is that the character of a matrix is not changed by any similarity transformation. From this it follows that the sum of the characters of the irreducible representations is equal to the character of the original reducible representation from which they are obtained. We have seen that for each symmetry operation the matrices of the irreducible representations stand along the diagonal of the matrix of the reducible representation, and the character is just the sum of the diagonal elements. When reducing a representation, the simplest way is to look for the combination of the irreducible representations of that group—that is, the sum of their characters in each class of the character table—that will produce the characters of the reducible representation. 

reduce the representation of the two N-H bond length changes of HNNH  

It may be asked, of course, whether this is the only way of decomposing the Ti representation. The answer is reassuring The decomposition of any reducible representation is unique. If we find a solution just by inspection of the character table, it will be the only one. Often this is the fastest and simplest way to decompose a reducible representation. 

A more general and more complicated way is to use a reduction formula  

The reduction formula can be simplified by grouping the equivalent operations into classes, 

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The dimension of a representation is the same as the order of the matrix. To reduce a representation it is necessary to reduce its order. It is noted that the dimension of a matrix representation corresponds to the character of the identity (E) matrix. 

Fig. 3.16 The efect of introducing a weak potential into the ID lattice is to lift the degeneracy of the energy levels mar to the edge of the Brillouin zone (shown in both extended-zone and reduced-zone representation).

Various techniques exist that make possible a normal mode analysis of all but the largest molecules. These techniques include methods that are based on perturbation methods, reduced basis representations, and the application of group theory for symmetrical oligomeric molecular assemblies. Approximate methods that can reduce the computational load by an order of magnitude also hold the promise of producing reasonable approximations to the methods using conventional force fields. 

The ratio PVC/CPVC (or reduced CPVC) is a simple but profound index of primer film structure that is perhaps the most important parameter in all of paint technology [70]. The relationship of the reduced CPVC to film structure can be appreciated by reference to Fig. 8. This figure shows a representation of a dry primer film with a range of pigment volume fraction that increases from left to right. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

FT is essentially a mathematical treatment of harmonic signals that resolved the information gathered in the time domain into a representation of the measured material property in the frequency domain, as a spectrum of harmonic components. If the response of the material was strictly linear, then the torque signal would be a simple sinusoid and the torque spectrum reduced to a single peak at the applied frequency, for instance 1 Hz, in the case of the experiments displayed in the figure. A nonlinear response is thus characterized by a number of additional peaks at odd multiples of the... 

These so-called Pareto-based techniques do not force consolidation over multiple criteria in advance and aim to return a representation of the set of optimal compounds. They support discussion between team members who may have different views on the downstream impacts of different risk factors perhaps, for example, one team member may know that there is a reliable biomarker for one potential side-effect. This would then mean that assessing this risk need not consume much development time and cost, and the risk factor can have a reduced weighting within the target product profile being evolved by the team. 

Since Lm is invariant under G, any operator A G transforms each vector >n Lm into another vector in Lm. Hence, the operation AM results in a matrix of the same form as T(A). It should be clear that the two sets of matrices I) 1) and D > give two new representations of dimensions m and n — m respectively for the group G. For there exists a set of basis vectors l, n] for rX2 The representation T is said to be reducible. It follows that the reducibility of a representation is linked to the existence of a proper invariant subspace in the full space. Only the subspace of the first m components is... 

The peculiarity of PCA is in a representation of the data set onto a subspace of reduced dimensionality where the statistical properties of the original data set are preserved. 

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

Products between the irreducible representation characters within a group will produce representations which are often reducible. A simple calculation can decompose this product to a sum of the irreducible representation characters, as is demonstrated in Table V for two representations from the S3-DP-S2 group. 

PCA is a data compression method that reduces a set of data collected on M variables over N samples to a simpler representation that uses a much fewer number (A M) of compressed variables , called principal components (or PCs). The mathematical model for the PCA method is provided below ... 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

Others (e.g., Fukashi Sasaki s upper bound on eigenvalues of 2-RDM [2]). Claude Garrod and Jerome Percus [3] formally wrote the necessary and sufficient A -representability conditions. Hans Kummer [4] provided a generalization to infinite spaces and a nice review. Independently, there were some clever practical attempts to reduce the three-body and four-body problems to a reduced two-body problem without realizing that they were actually touching the variational 2-RDM method Fritz Bopp [5] was very successful for three-electron atoms and Richard Hall and H. Post [6] for three-nucleon nuclei (if assuming a fully attractive nucleon-nucleon potential). 

When N = p, the set B simply contains the p-particle reduced Hamiltonians, which are positive semidefinite, but when N = p + 1, because the lifting process raises the lowest eigenvalue of the reduced Hamiltonian, the set also contains p-particle reduced Hamiltonians that are lifted to positive semidefinite matrices. Consequently, the number of Wrepresentability constraints must increase with N, that is, B C B. To constrain the p-RDMs, we do not actually need to consider all pB in B, but only the members of the convex set B, which are extreme A member of a convex set is extreme if and only if it cannot be expressed as a positively weighted ensemble of other members of the set (i.e., the extreme points of a square are the four corners while every point on the boundary of a circle is extreme). These extreme constraints form a necessary and sufficient set of A-representability conditions for the p-RDM [18, 41, 42], which we can formally express as... 

The three complementary representations of the reduced Hamiltonian offer a framework for understanding the D-, the Q-, and the G-positivity conditions for the 2-RDM. Each positivity condition, like the conditions in the one-particle case, correspond to including a different class of two-particle reduced Hamiltonians in the A-representability constraints of Eq. (50). The positivity of arises from employing all positive semidefinite in Eq. (50) while the Q- and the G-conditions arise from positive semidefinite and B, respectively. To understand these positivity conditions in the particle (or D-matrix) representation, we define the D-form of the reduced Hamiltonian in terms of the Q- and the G-representations ... 

J. R. Hammond and D. A. Mazziotti, Variational two-electron reduced-density-matrix theory partial 3-positivity conditions for A-representability. Phys. Rev. A 71, 062503 (2005). 

Nakatsuji [37] in 1976 first proved that with the assumption of N-representability [3] a 2-RDM and a 4-RDM will satisfy the CSE if and only if they correspond to an A-particle wavefunction that satishes the corresponding Schrodinger equation. Just as the Schrodinger equation describes the relationship between the iV-particle Hamiltonian and its wavefunction (or density matrix D), the CSE connects the two-particle reduced Hamiltonian and the 2-RDM. However, because the CSE depends on not only the 2-RDM but also the 3- and 4-RDMs, it cannot be solved for the 2-RDM without additional constraints. Two additional types of constraints are required (i) formulas for building the 3- and 4-RDMs from the 2-RDM by a process known as reconstruction, and (ii) constraints on the A-representability of the 2-RDM, which are applied in a process known as purification. 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

J. E. Harriman, Geometry of density matrices. 2. Reduced density matrices and A-representability. Phys. Rev. A 17, 1257 (1978). 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

The remarkable fact, first demonstrated by Nakatsuji [18], is that for each p >2, CSE(p) is equivalent (in a necessary and sufficient sense) to the original Hilbert-space eigenvalue equation, Eq. (2), provided that CSE(p) is solved subject to boundary conditions (A -representability conditions) appropriate for the (p + 2)-RDM. CSE(p), in other words, is a closed equation for the (p+ 2)-RDM (which determines the (p + 1)- and p-RDMs by partial trace) and has a unique A -representable solution Dp+2 for each electronic state, including excited states. Without A -representability constraints, however, this equation has many spurious solutions [48, 49]. CSE(2) is the most tractable reduced equation that is still equivalent to the original Hilbert-space equation, and ultimately it is CSE(2) that we wish to solve. Importantly, we do not wish to solve CSE(2) for... 

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