Operations algebraic expressions

For cases in which the equiUbtium and operating lines may be assumed linear, having slopes E /and respectively, an algebraic expression for the integral of equation 55 has been developed (41) ... 

Another loose end is the relationship between the quasi-algebraic expressions that matrix operations are normally written in and the computations that are used to implement those relationships. The computations themselves have been covered at some length in the previous two chapters [1, 2], To relate these to the quasi-algebraic operations that matrices are subject to, let us look at those operations a bit more closely. 

Here Et is the total enzyme, namely, the free enzyme E plus enzyme-substrate complex ES. The equation holds only at substrate saturation, that is, when the substrate concentration is high enough that essentially all of the enzyme has been converted into the intermediate ES. The process is first order in enzyme but is zero order in substrate. The rate constant k is a measure of the speed at which the enzyme operates. When the concentration [E]t is given in moles per liter of active sites (actual molar concentration multiplied by the number of active sites per mole) the constant k is known as the turnover number, the molecular activity, or kcat. The symbol fccat is also used in place of k in Eq. 9-6 for complex rate expressions in which fccat cannot represent a single rate constant but is an algebraic expression that contains a number of different constants. 

Transcendental functions are mathematical functions which cannot be specified in terms of a simple algebraic expression involving a finite number of elementary operations (+,... 

In the general case, there is no simple algebraic expression for the first term of (5.42), therefore, for its evaluation one must utilize the tables of numerical values of the submatrix elements of operators Uk. The most complete tables, covering also the case of operators Vkl, may be found in [87], For operators, also depending on spin variables, the analogue of formula (5.41) will have the form... 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

In [90] the relationship between eigenvalues of the Casimir operators of higher-rank groups and quantum numbers v, N, L, S is taken into account to work out algebraic expressions for some of the reduced matrix elements of operators (Uk Uk) and (Vkl Vkl). However, the above formulas directly relate the operators concerned, and some of these formulas are not defined by the Casimir operators of respective groups. 

The method of CFP is an elegant tool for the construction of wave functions of many-electron systems and the establishment of expressions for matrix elements of operators corresponding to physical quantities. Its major drawback is the need for numerical tables of CFP, normally computed by the recurrence method, and the presence in the matrix elements of multiple sums with respect to quantum numbers of states that are not involved directly in the physical problem under consideration. An essential breakthrough in this respect may be finding algebraic expressions for the CFP and for the matrix elements of the operators of physical quantities. For the latter, in a number of special cases, this can be done using the eigenvalues of the Casimir operators [90], however, it would be better to have sufficiently simple but universal formulas for the CFP themselves. 

Equations of this kind can also be derived for the special cases of reduced matrix elements of operators composed of irreducible tensors. Then, using the relation between CFP and the submatrix element of irreducible tensorial operators established in [105], we can obtain several algebraic expressions for two-electron CFP [92]. Unfortunately such algebraic expressions for CFP do not embrace all the required values even for the pN shell, which imposes constraints on their practical uses by preventing analytic summation of the matrix elements of operators of physical quantities. It has turned out, however, that there exist more general and effective methods to establish algebraic expressions for CFP, which do not feature the above-mentioned disadvantages. 

Similar expressions can also be obtained at N = 4 [108]. To establish similar algebraic expressions for the repeating terms vLS only requires that the appropriate second-quantized operators (16.7), (16.8), (16.10) and (16.11) be used. For example, instead of (16.63), we have... 

The above recurrence relations allow the numerical values of CFP to be readily found and the availability of the above algebraic formulas for them makes it possible to establish similar algebraic expressions for other quantities in atomic theory, including the CFP with two detached electrons, the submatrix elements of irreducible tensors, and also the matrix elements of the operators of physical quantities [110]. 

Making use of the properties of the eigenvalues of Casimir operators, mentioned in Chapter 5, we are in a position to find a number of interesting features of the matrix elements of the Coulomb interaction operator. Thus, it has turned out that for the pN shell there exists an extremely simple algebraic expression for this matrix element... 

Again, using the properties of the Casimir operators, we can establish the following simple algebraic expressions for these corrections (0 <, N < 41 + 2, for s-electrons the orbit-orbit interaction vanishes) ... 

For the fN shell we have to take into consideration terms containing expressions of the kind (5.36) and (5.37). As was shown in [127, 129], parameters a and / account for the superposition of all configurations which differ from ground lN by two electrons. If the admixed configurations differ from the principal one by the excitation of one electron, then we have to introduce one extra parameter T, the coefficient of which will be described by the matrix element of the tensorial operator of the type [Uk x Uk x Uk ]°, for which, unfortunately, there is no known simple algebraic expression. This correction is important only for N > 3. 

During the last two decades a number of new versions of the Racah algebra or its improvements have been suggested [27]. So, the exploitation of the community of transformation properties of irreducible tensors and wave functions allows one to adopt the notion of irreducible tensorial sets, to deduce new relationships between the quantities considered, to simplify further on the operators already expressed in terms of irreducible tensors, or, in general, to offer a new method of calculating the matrix elements, as an alternative to the standard Racah way. It is based on the utilization of tensorial products of the irreducible operators and wave functions, also considered as irreducible tensors. 

Chemists use the p-function operator to express the concentrations of many ions. pOH for hydroxide ion, pCa for Ca2+, and so forth. The meaning of the p is the same in every case—take the logarithm of the concentration and then change the algebraic sign. 

The present choice can be algebraically expressed by the following definition of the rotation operator... 

As we shall see, it is very helpful to be able to compute the behavior of the spin system from the sort of matrix multiplications that we have already carried out. On the other hand, it is often possible to simplify the algebraic expressions by using the corresponding spin operators. In fact, this is the concept of the product operator formalism that we discuss later. Note that from Eqs. 11.35 and 11.36, the (redefined) density matrix at equilibrium can be written in operator form as... 

Symmetry operations may be represented by algebraic equations. The position of a point (an atom of a molecule) in a Cartesian coordinate system is described by the vector r with the components x, y, z. A symmetry operation produces a new vector r with the components t, /, z. The algebraic expression representing a symmetry operation is a matrix. A symmetry operation is represented by matrix multiplication. 

This diagram contains two pairs of equivalent lines—that is, lines beginning at the same operator interaction line and ending at the same interaction line. For each such pair, a prefactor of Vi is multiplied onto the algebraic expression. ... 

The permutation operators appear in order to maintain the antisymmetry of the algebraic expressions, as explained earlier. Note that the factors of lA appearing in the second and third terms result from both a pair of equivalent lines and a pair of equivalent vertices in each of the corresponding diagrams. 

Rule 4. Generate the algebraic expressions corresponding to the non-canonical diagrams directly from the algebraic expressions for the canonical diagrams by applying permutation operators. 

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