Double-counting

The factor of 2 in the first tenn comes from spin. In the exchange temi, there is no extra factor of 2 because one can subtract off a double-counting tenn (see equation Al.3.19). The simnnations can be executed as per equation A 1.3.29 to yield... 

These two expressions differ in a very important way the sum of occupied orbital energies double counts the Coulomb minus exchange interaction energies. Thus, within the FIE approximation, the sum of the occupied orbital energies is not equal to the total energy. 

Thus Ek (the sum of orbital energies) and Ek< l (the correction for double counting) add up to produce the proper expectation value energy. 

Adjustment required to eliminate double counting of people. 

The reuse or recycling of wastes can reduce the amount of freshwater and raw materials required for a process. While looking at the inputs to unit operations, think about the opportunities for reusing and recycling outputs from other operations. Note that if reused wastes are not properly documented, double-counting may occur in the material balance, particularly at the process or complete plant level that is, a waste will be quantified as an output from one process and as an input to another. 

The value of production attributable to a particular establishment may be adjusted by subtracting the value of products obtained from other establishments within the same facility that are incorporated into its final products. This procedure eliminates the potential for double counting" production in situations where establishments are engaged In sequential production activities at a single facility. 

This is an important property when converting to probability to avoid double counting. 

Incestuous data Data in two or more data sets that are derived from a common origin and may be inadvertendy double-counted when aggregated. 

The on-site energy Dp for atomic species Q at the lattice site R consists of the band, core, double-counting, non-spherical and Madelung contributions, respectively. 

Nj is the number of d electrons of the atom at site i and the second term in the bracket avoids the double counting of the change in the electron-electron interactions that are responsible for the shift of the atomic level AVi. C is the set of atoms significantly perturbed by the defect and AErep is the perturbation of the pair wise repulsive ir aractions. 

Two further consequences of the steric activity of open d shells are also important. One, which might seem somewhat circular but does not, in fact, involve any double counting , is that longer bonds are accompanied by smaller ligand-fields, that is, by... 

Households have direct and indirect costs. Direct household costs imply payments for transport to and from the health services, acconunodation for the accompanying relative, special buildings for disabled patients (e.g., changed bathroom), costs for diet, and the re-education, for instance new training after a paralysis. Private households also have to bear direct payments for user fees and drugs, which are an income of the providers. There must be clearing between these components of the COI to avoid double-counting. 

The effect of double counting is most easily seen in the following calculation. Suppose that the density of molecules is Pa = Pb = 10 and that A and B are identical. Consequently, the number of collisions between A and B is... 

As mentioned, xA(r) is half of the total adiabatic change in the subsystem macrostate associated with the current phase space point T. The factor of is used to compensate for double counting of the past and future changes. In the steady state, the subsystem most likely does not change macrostate, and hence this change has to be compensated by the change in the reservoir, Axr = xA(r). 

ONIOM can combine two MO levels like ONIOM(QM QM), which is a unique feature that is not available to QM/MM methods. However, the most popular combination is ONIOM(QM MM), combining QM with MM. This is essentially equivalent to generic QM/MM. However, there are some subtle cancellation or double-counting differences for the case where a covalent bond is cut. For this, we refer to a detailed discussion published elsewhere [8], QM MM or QM/MM applications have typically been used without appropriate accuracy or S-value tests, as the benchmark full QM calculation for the real system is often impossible. In Section 2.2.2, we will examine one such test in detail. 

Heavy cmde oil will not be explicitly addressed, as there are often no clear boundaries with conventional oil and extra-heavy crude oil within a heavy oil reservoir. To avoid double counting, reserves of heavy oil will not be considered either, as they are already partially or completely included in conventional or extra-heavy oil reserves (BGR, 2003). 

Here the 2 in the denominator is to avoid double counting because it takes two sites to form one link. However closed loops and chain entanglements are both possibilities and Equation (2.57) must be modified for these effects. We can write the network modulus as... 

However, in all cases care has to be taken to avoid double-counting various permutations. A good correction (Kikuchi 1977) is to divide by a factor which numerically describes the nearest-neighbour (n) connectivity (c) of the lattice, which for pairs leads to... 

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