Derived quantities

Essentially, the results of a quantum mechanical calculation by the SCF or Htickel methods are the eigenvectors and the corresponding orbital energies, as well as the total energy. Configuration interaction yields, in addition, the coefficients of the various configurations and the modified total energies. These results, as such, cannot be compared directly with experimental data. Their interpretation requires the evaluation of additional quantities. 

The electron density defined by a MO can be decomposed into atomic and bond contributions. 

The total electron density at an atom t is evaluated as the sum of the electron densities contributed by each electron in each qn, i.e., 

The total bond order puv for the bond between atoms and v is defined as 

It is assumed that the bond order gives an indication of the strength for the corresponding bond. 

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The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

The most commonly used system apart from SI is the cgs system based on the =ntiinetre, gram and second as the only base units. The unit of force is the dyne, ind the unit of energy is the erg. In electromagnetism, SI is associated with an independent base quantity of current, whereas cgs is associated with current as a derived quantity. 

Here (3A — Nc) is the number of degrees of freedom, equal to three times the number of particles minus the number of constraints, which typically will be 3 (corresponding to conservation of linear momentum). In a standard MC simulation the temperature is fixed NVT conditions), while it is a derived quantity in a standard MD simulation NVE conditions). 

Results of scientific observations are often combined. For example, in Experiment 5 you will determine the change of water temperature during the combustion of a candle (or during the solidification of candle wax). The change of temperature, which we called At, is the result of two measurements, not just one—it is a derived quantity ... 

We see that the uncertainty in a derived quantity is fixed by the uncertainties in the measurements that must be combined. For an addition or a subtraction, the maximum uncertainty is simply the sum of the uncertainties in the components 0.2 + 0.2 = 0.4. 

Once again the uncertainty in the product, a derived quantity, is fixed by the uncertainties in the measurements that must be combined. 

We need convenient rules for estimating the maximum uncertainty in derived quantities. This is rather easy for a sum or a difference. In either case, merely add up the uncertainties in the components. Fortunately there is an equally sim-... 

The energy values and the derived quantities for the diatomic molecules of the alkali metals are given in table 1. It is seen that the amount of p character is calculated to he between 5 and 14 %. 

In these equations, J is the physical current density. Clearly, the magnetisation density is a derived quantity. 

The concepts of ionic and covalent character of a bond are vague and ill defined. The well-defined AIM-derived quantities such as the integrated atomic charges and the bond critical point density provide a quantitative characterization of bonding (4). 

In the framework of the force field calculations described here we work with potential constants and Cartesian coordinates. The analytical form of the expression for the potential energy may be anything that seems physically reasonable and may involve as many constants as are deemed feasible. The force constants are now derived quantities with the following definition expressed in Cartesian coordinates (x ) ... 

The Dq and B values reported for hexachloro species of the 4 d and 5 d series therefore given in Table 25, together with those of various derived quantities. In the Table the effective metal t values are also given, where... 

Subroutine PLOTTER actually plots the relative values of the elements of an array called ploty. In this subroutine it is necessary only to set the values of ploty according to the labels specified in GRAFINIT. It is not necessary to plot all of the dependent variables, y, and it is possible to plot derived quantities like the concentrations of dissolved carbon species, by simply specifying what is wanted in GRAFINIT and PLOTTER. The cores of these two subroutines, GRINC and PLTC, do not need to be modified for different simulations. 

For the silyl compounds (Table 27) high-quality calculated BDE values have also been included with the experimentally derived quantities. Here, the uncertainties and distribution of experimental values are larger than for the methyl compounds. The calculated BDEs in this work and corresponding comparison experience with the methyl compounds can, perhaps, be used to narrow some of the uncertainties in the silyl compounds. For almost all the germaniun, tin and lead compounds in Tables 28-30, the calculated BDEs are the only information available to date. 

Density is a derived quantity combining mass and volume. It is defined as mass per unit volume of a substance at a fixed temperature and pressure. It... 

As referred to in the previous chapter, in bomb combustion calorimetry the reaction proceeds inside a pressure vessel—the bomb—at constant volume, and in this case the derived quantity is Ac U°. In flame calorimetry the reaction occurs in a combustion chamber, which is in communication with the atmosphere, and the measurements lead to ACH°. The methods of combustion calorimetry will be described in the following paragraphs. 

This idea of current flowing as a function of polarizing the electrode (shifting its potential away from equilibrium) lies at the very heart of voltammetry. Note that the magnitude of the current - and current is a derivative quantity (equation (2.1)) - tells us the rate at which the electrochemical conversion occurs. 

The domain in which GEORGE operates is a small but important one for introductory chemistry. He works with problems involving the fundamental quantities mass, volume, and number of moles. He can also work with derived quantities such as density, molar mass, molar... 

The reaction probability model produces a complete calculated C nmr spectrum as a best fit to the observed experimental spectrum. The deduced probabilities provide the following derived quantities (1) "rira", a measure of monomer sequence randomness, (2) the distribution of methylene sequence lengths,... 

The SI consists of seven base quantities from which all the other quantities (secondary or derived quantities) can be derived. The table in the slide presents the name of the quantities, their unit and the symbol of the unit. 

Some examples of derived quantities are given in this shde. In particular the last one is widely used in analytical chemistry. 

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