Atoms higher-order approximations

Let us take, for instance, the value of N in NO2 and NH2. If different values are used for N in the two groups, then a similar higher order approximation should also be used for other atoms, such as O in COOH and CO. 

The Electronic States of Atoms. III. Higher-Order Approximations... 

Nevertheless, the examination of the applicability of the crude BO approximation can start now because we have worked out basic methods to compute the matrix elements. With the advances in the capacity of computers, the test of these methods can be done in lower and lower cost. In this work, we have obtained the formulas and shown their applications for the simple cases, but workers interested in using these matrix elements in their work would find that it is not difficult to extend our results to higher order derivatives of Coulomb interaction, or the cases of more-than-two-atom molecules. 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

The anharmonicity of atomic vibration in crystals may also be approximated by adding higher-order terms to the potential energy function of equation 3.2 ... 

Consider the exact definition of from Eq. (10.32). When atom fe is a sp carbon, we can safely neglect the second- and higher-order terms because the values are small, in favor of the simple approximation, Eq. (10.41). However, we must consider both (T- and rr-electron densities and their variations. The appropriate first derivatives dEf"/dNk)° are indicated in Table 10.3. 

We will carry out our program in two steps. In this section we will derive the two-particle density operator Fn in a three-particle collision approximation for the application in the collision integral of Fl. As compared with Section II.2, the main difference will be the occurrence of bound states and, especially, the generalization of the asymptotic condition, which now has to account for bound states too. For the purpose of the application in the kinetic equation of the atoms (bound states) we need an approximation of the next-higher-order density matrix, that is, F 23 This quantity will be determined under inclusion of certain four-particle interaction. 

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