Direct sums

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

The internal (potential) energy is a direct sum of energies, which is normally given as a sum over pairwise interactions (i.e. van der Waals and electrostatic contributions in a force field description). 

In order to understand the structure of this norm a little better, it is appropriate to introduce the space H = H H being the direct sum of two copies of H. The space H is defined as the set of all vectors of the form... 

This direct sum with respect to an index which is common to the creator and the corresponding annihilator cannot always be carried out as will be seen in the following sections, hence the usefulness of the general MCM (relation (13)). 

Inequation (18) the D(j k) terms are n-electron Slater determinants formed by the spin-orbitals numbered by means of the direct sum j0k of the vector index parameters attached to the involved nested sums and to the occupied-unoccupied orbitals respectively. That is ... 

In addition one can always find a transformation leading to a symmetry adapted basis [4] e, so that T is brought to the block diagonal form T via the associated similarity transformation. The T matrix can be written as a direct sum... 

It should be apparent that the summation indicated in Eq. (28) is special. It indicates that the direct sum is to be taken that is, the various irreducible representations are arranged in arbitrary order along the diagonal of the reduced... 

Here, I as given by the direct sum, is a (reducible) representation of a given operation, R, Its trace is the character, a quantity that is independent of tire choice of basis coordinates. As xr is merely the sum of the diagonal elements of T, it is also equal to the sum of the traces of the individual submatrices... 

Although the resulting direct product may not be reduced, it can be made so by application of the magic formula, or often by inspection. The nonvanishing of the integral is then determined by the existence of the totally symmetric representation in the resulting direct sum. This procedure will be illustrated by the development of spectroscopic selection rules in Section 12.3.3. 

The right-hand side of Eq. (96) is of course the weighted direct sum of the irreducible representations. By convention the totally symmetric irreducible representation corresponds to t = 1. Thus, if n(1> = 0, the integral in Eq. (95) vanishes. The transitions m -> nandm n are then forbidden by the symmetry selection rules. Thte principle can be illustrated by the following example. 

The internal coordinates for the water molecule are chosen as changes in the structural parameters defined in Fig. 3. The effect of each symmetry operation of the symmetry group ( 2 on these internal coordinates is specified in Table 2. Clearly, the internal coordinate Ace is totally symmetric, as the characters xy(Aa) correspond to those given for the irreducible representation (IR) Ai. On die other hand, the characters x/(Ar), as shown, can not be identified with a specific IR. By inspection of Table 2, however, it is apparent that the direct sum Ai B2 corresponds to the correct symmetry of these coordinates. In more complicated cases the magic formula can always be employed to achieve the correct reduction of the representation in question. 

The saddle-point equation leads to the momentum dependent dynamical quark mass Mf(k) = MfF2(k). Mf here is a function of current mass mf (M.M. Musakhanov, 2002). It was found that that M[m] is a decreasing function and for the strange quark with ms = 0.15 GeV Ms 0.5 Mu>d. This result in a good correspondence with (P. Pobylitsa, 1989), where another method was completely applied - direct sum is of planar diagrams. 

In this case, it can be proved that the canonical SCF orbitals, being solutions of Eq. (26), are symmetry orbitals, i.e. that they belong to irreducible representations of the symmetry group. 12) If the number of molecular orbitals is larger than the dimension of the largest irreducible representation of the symmetry group, it must then be concluded that the set of all N molecular orbitals form a reducible representation of the group which is the direct sum of all the irreducible representations spanned by the CMO s. 

Since the operators b]a, bia with different indices are assumed to commute, the algebraic structure of many-body quantum mechanics is the direct sum of the algebras of each degree of freedom... 

Some algebras have identical commutation relations. They are therefore called isomorphic algebras. A list of isomorphic algebras (of low order) is shown in Table A.3. In this table, the sign denotes direct sums of the algebra, that is, addition of the corresponding operators. There is also the trivial case U(l) SO(2). 

The simplest example of coupled algebras is provided by the angular momenta (cf. note 2 of Chapter 1). The direct sum of two angular momenta, J and J2, is just... 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

The Fock-space Hamiltonian H is equivalent to the configuration-space Hamiltonian H insofar as both have the same matrix elements between n-electron Slater determinants. The main difference is that H has eigenstates of arbitrary particle number n it is, in a way, the direct sum of aU // . Another difference, of course, is that H is defined independently of a basis and hence does not depend on the dimension of the latter. One can also define a basis-independent Fock-space Hamiltonian H, in terms of field operators [11], but this is not very convenient for our purposes. 

As in the previous section we consider a single Slater determinant reference function with the spin orbitals i/, occupied. However, we express our excitation operators in a completely arbitrary basis of spin orbitals i/, which is no longer the direct sum of occupied and unoccupied spin orbitals. Then the following replacements must be made [3] ... 

This means that these D(4) matrices are really a combination of two separate group representations (mathematically, it is called a direct sum representation . We say that D 4) is reducible into a one-dimensional representation D 4) and a three-dimensional representation formed by the 3x3 submatrices which we will call D(3). 

Thus, our original Dd) representation was a combination of two Aj representations and one E representation. We say that Dd) is a direct sum representation Dd) = 2Ai E. A consequence is that the characters of the combination representation Dd) can be obtained by adding the characters of its constituent irreducible representations. 

One considers a Hilbert space that can be fragmented as the direct sum of three subspaces H = Qi Q2 The fragmentation of the Hilbert space in three... 

One way to combine vector spaces is to take a Cartesian sum. (Mathematicians sometimes call this a Cartesian product. Another common term is direct sum.)... 

A reducible representation is said to be the direct sum of the irreducible representations of which it is made up. In the standard notation (Section 9.5) for point-group irreducible representations, the 03v representation (9.29) is called A ( and the representation (9.28) is called E. If we denote the reducible representation (9.25) by T, then... 

Since each MO belongs to some irreducible representation of the molecular point group, we must find linear combinations of the AOs that transform according to the irreducible representations group theory enables us to do this. A symmetry operation sends each nucleus either into itself or into an atom of the same type a symmetry operation will thus transform each AO into some linear combination of the AOs (9.63). Therefore (Section 9.6), the AOs form a basis for some representation TAO of the point group of the molecule. This representation (as any reducible representation) will be the direct sum of certain of the irreducible representations r r2,...,r (not necessarily all different) of the molecular point group ... 

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