Products of Operations

To see exactly the result of successive operations, the end point of one operation is used as the start point of the next. Some products are easy to visualize for example, the product C2 C2 i.e. a rotation by 360°, leaves the molecule unchanged and in exactly the same configuration as the starting point. If we are to have a closed group, then this combined operation must be the result of a single operation in the group, and so a new operation is introduced, the identity operation E. 

On application of the identity operation E, nothing moves at all so the entire molecule is the symmetry element. No matter what the shape of a molecule, a rotation of 360° about any axis would give an indistinguishable arrangement of the atoms. This is because a rotation of 360° is equivalent to doing nothing to the molecule. It does not matter what axis is used, or even if the operation is done at all, and so all molecules possess the identity symmetry element. We will see that other repeated applications of other symmetry operations can also lead to the case of an identical molecule. 

For more complex sequences of operations it is helpful to use drawings or models including the vectors to understand the results of successive operations. Appendix 1 gives some templates for paper models of water illustrating the vectors used here for each end 

To identify all the possible products systematically, a multiplication table of symmetry operations can be drawn up in which each row and column of the table has one of the symmetry operations as a heading the body of the table then contains the operation resulting from the product of that row and column. The starting point for the case of HjO is given in Table 2.1 and it is left to the reader to complete this following the instructions in Problem 2.1. 

Problem 2.1 The results of the operation products for Table 2.1 have been left blank. Work out the single operation equivalent to each product and fill in the table. The paper models from Appendix 1 can be used to help in this exercise. Start with the model representing the result of the first operation and then carry out the second. Compare the configuration obtained with the three other models to identify the end point. You should find that for every pair of operations there is always a model that looks the same as the end point you have come up with. A completed table is included at the end of the chapter. 

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For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

The opposite of a creation operator is an annihilation operator a which removes orbital i from the wave function it is acting on. The a-a product of operators removes orbital j and creates orbital i, i.e. replaces the occupied orbital j with an unoccupied orbital i. The antisymmetry of the wave function is built into the operators as they obey the following anti-commutation relationships. 

Practical activities should embody as best as possible the scientifie proeesses that have been preseribed by the American Association for the Advancement of Science observation, elassification, numerieal relations, measurements, time-spaee relations, eommunieation (oral, pictorial, written), deriving of conclusions, prediction ( what would happen if. .hypothesis making, production of operational definitions, identifieation and control of variables, experiment and explanation of experimental data. Different theoretical perspectives should be used with the aim to optimize the positive eognitive and affeetive outcomes. The use, sometimes together, sometimes separately, of different perspeetives can act complimentarily and can lead to positive results (Niaz, 1993 Tsaparhs, 1997). 

The resulting matrix F corresponds to the 4/3 power of the original matrix R. If more advanced, GGA-type functionals are used rather than the local density approximation, the procedure becomes slightly more complicated due to the more complex forms of the functionals. Here we just briefly sketch the general strategy which is centered around the observation that these functionals can usually be interpreted as a product of operators con-... 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

When evaluating matrix elements of operators for coupled systems, it is often convenient to make use of reduction formulas. These formulas reduce the evaluation of products of operators to matrix elements of individual operators. Two situations can occur. 

Expectation values of products of operators with respect to are simply the sum of all full contractions, for example. 

Eor products of operators a in the generalized normal order we then get... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

The Casimir operator of (21 + l)-dimensional orthogonal group R21+1 may be expressed in terms of the sum of scalar products of operators Uk... 

If in the irreducible tensorial product of operators (14.40) and (14.42) we interchange the second-quantization operators connected with an arrow,... 

If the specific tensorial structure (14.58) of a two-electron operator is known, then we can obtain its representation in terms of the product of operators (14.30) acting in the space of states of one shell. In fact, if we substitute into the two-electron matrix element which enters into (13.23), the operator (14.58) in the form... 

Let us now return to the Casimir operators for groups Spy+2, SU21+1, R21+1, which can also be expressed in terms of linear combinations of irreducible tensorial products of triple tensors WiKkK To this end, we insert into the scalar products of operators Uk (or Vkl), their expressions in terms of triple tensors (15.60) and then expand the direct product in terms of irreducible components in quasispin space. As a result, we arrive at... 

Exercise 2.1-4 Construct the multiplication table for the set E Sf C2 S4. Demonstrate by a sufficient number of examples that this set is a group. [Hint Generally the use of projection diagrams is an excellent method of generating products of operators and of demonstrating closure.] In this instance, the projection diagram for S4 has already been developed (see Figure 2.8). 

Wick s theorem (35) which gives us the prescription for treating a product of operators may of course be applied to the Hamiltonian, expressed in the second quantization formalism (29). This leads to the Hamiltonian in a form which is of primary importance in perturbation treatments. This form of the Hamiltonian which is called the normal product form is ... 

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... 

A somewhat more general version of Wick s theorem may be developed which involves products of operator strings, some or all of which may be normal-ordered. The original form of Wick s theorem is only slightly modified in that the contractions need be evaluated only between normal-ordered strings and not within them. For example, for a product of two normal-ordered strings, the generalized Wick s theorem says that... 

It is also easily demonstrated that operator equations for sums and products of operators are cast into equivalent expressions by a unitary transformation. 

This algorithm can be implemented in a typical program using the identity exp a d/dg x))]x = g " g x) + a] and the direct translation technique [29]. Here, the product of operators from (97) are translated into a set of instructions which alleviate the need to calculate analytically the phase space vector r At) in terms of the initial conditions T(0). A pseudocode for performing these operations would appear as ... 

To proceed, we must evaluate the Wigner transform of a product of operators. This calculation is given in several reviews [2] but we sketch it here for completeness. Letting Q = Tip- Zj2 and Q = R — Zj2, we may invert the relation (4) to obtain... 

The partial Wigner transform of a product of operators satisfies the associative product rule,... 

Since both the particle and hole labels are in general permuted, and it is often the case that the permutations are independent, it is efficacious to (always) write the permutation operator as the binary product of operators acting on the disjoint sets,... 

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