Spherically symmetric function

In essence, it is the probability density of the two nuclei to have relative separation Since the orientation of the molecule is not fixed (nuclei are not fixed any more if we deal with an non-BO approach), gi( ) is a spherically symmetric function. The plots of gi ( ) are presented in Figs. 1-4-. It should be noted that all the correlation functions shown are normalized in such a way that... 

This normalization implies that a population parameter equal to 1 corresponds to a population of one electron for the spherically symmetric function d00. The nonspherical functions (/ > 0) represent a shift of density between regions of opposite sign. They have both positive and negative lobes which integrate to equal but opposite numbers of electrons. For these functions, the normalization expression... 

Proof. Let V denote the set of solutions in 2(]R3) obtained by multiplying a spherical harmonic by a spherically symmetric function ... 

While at the National Resource for Computation in Chemistry, I have developed a general classical simulation program, called, CLAMPS (for classical many particle simulator)(5) capable of performing MC and MD simulations of arbitrary mixtures of single atoms. The potential energy of a configuration of N atoms at positions R = (r, ..., r and with chemical species aj,..., o is assumed to be a pairwise sum of spherically symmetric functions. 

As V reduces to this form if is a spherically symmetrical function, it follows that spherically symmetrical solutions must satisfy the equation... 

Here Yio( ) oc cos 9 is the spherical harmonic with l = 1 and m = 0 that is obtained when a spherically symmetric function is differentiated with respect to 2. Expanding the potential (r) in the organic material, where there are no charges, in terms of multipoles, we see that only the dipole term, which is proportional to Yio( ), will be different from zero. Therefore the potential (r) in the organic medium is formally identical to the potential of an effective point dipole with dipole moment peff ... 

Our notation here essentially follows Refs. 4 and 5, which in turn closely followed that of Lebowitz, Stell, and Baer and of Stell, Lebowitz, Baer, and Theumann. There are minor differences from paper to paper, however. Our 2 and W here are the 2 and U fVof Ref. 4. In Ref. 6 n W is used to denote what we call IT here, whereas in Ref. 7, W is used to denote our 2. The 13(12) here is the <1> of Refs. 6 and 7. Moreover, the modified two-particle functions we denote here with the subscript S are denoted in the papers above with a caret, and their Fourier transforms carry a bar. (We reserve the caret and bar here to denote certain spherically symmetric functions and Hankel transforms that play a fundamental role in the mathematics of polar fluids.) Finally, in Refs. 4 and 5, p(l), p(12), and F 2) were written as p,(l), P2(12), and Fjfn), respectively. The subscripts are redundant when one exhibits the arguments, so we drop them here. 

The flD(r) can be regarded as the Fourier transforms of certain spherically symmetric functions dp r), which were introduced by Wertheim in his analysis of the mean spherical approximation for dipolar spheres. If a(r)- 0... 

This represents the 47t steradians of solid angle, which radiate from every point in three-dimensional space. For integration over a spherical symmetrical function F(r), independent of 6 and (p, Eq. (10.40) can be simplified to... 

The one-electron atom in the ground state has quantum numbers n = 1,1 = 0, mi = 0. The probability density depends on neither 6 nor cp, and therefore the probability density is a spherically symmetrical function. For other states the probability density in the plane perpendicular to the z axis becomes more and more pronounced as I increases from 0 to 4 (Figure 3.4). 

But, K is a spherically symmetric function in velocity space hence,... 

Since this is a function of r only, it is a spherically symmetric function. (In fact, all s orbitals are spherically symmetric.) The 2s orbital is more spread out than the Is orbital because the exponential in 2s decays more slowly and because the exponential is multiplied by Zr/ao (the 2 becomes negligible con iared to Zr/ao at large r). As a result, the charge cloud associated with the 2s orbital is more diffuse. (For this reason, when we approximate a polyelectronic atom like beryllium by putting electrons in Is and 2s orbitals the 2s electrons are referred to as outer and the Is electrons are called iimer. )... 

A complication arises for functions of d or higher symmetry. There are five real d orbitals, which transform as xy, xz, yz, x —y, and z, that are called pure d functions. The orbital commonly referred to as is actually Iz —x —y. An alternative scheme for the sake of fast integral evaluation is to use the six Cartesian orbitals, which are xy, xz, yz, x, y, and These six orbitals are equivalent to the five pure d fimctions plus one additional spherically symmetric function x +y +z ). Calculations using the six d functions often yield a very slightly lower energy due to this additional function. Some ab initio programs give options to control which method is used, such as Sd, 6d, pure-J, or Cartesian. Pure- f is equivalent to 5d and Cartesian is equivalent to 6d. Similarly, If and 10/ are equivalent to pure-/ and Cartesian / functions, respectively. 

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