Nodal circle

All of these densities feature two points in the zz-plane where the density vanishes exactly. They are situated on the x-axis, as sections along that axis demonstrate clearly. We show those in Fig. (2). Independently of the mixing coefficient a, those nodal points occur at x = 1/2 on each equatorial axis. They are the intersection of the aforementioned nodal circle with the displayed plane. 

This significant m variability is reduced but does not disappear if we switch to the real spherical harmonics, which are arguably more important chemically, particularly in minimal-basis-set applications. In the context of chemistry we almost never use the complex spherical harmonics but rather their real and imaginary parts. The real spherical harmonics each have / nodal circles... 

The next concept we need is the nodal circle. If all paths (constructed from a sequence of bonds) from circle 1 to circle 2 pass through the same black circle, then that black circle is called a nodal circle. If one nodal circle is removed, the graph falls into two disconnected pieces, both of which contain a white circle. A graph may have any number of nodal circles from zero upwards. As an example, the following graph with white circles... 

C2(l,2) = The sum of all distinct connected simple graphs consisting of 2 white 1-circles, some or no pi-circles and, /-bonds such that the graphs are free of nodal circles and articulation circles... 

Chandler and his co-workers have taken a somewhat different point of view in seeking to improve upon the SSOZ equation. They start from the perspective that the integral equation itself is flawed since all the tractable closures correspond to the resummation of unallowed diagrams in the interaction site cluster series for the site-site total correlation function. They have formulated an integral equation in which the direct correlation function does indeed correspond to the subset of diagrams in the interaction site cluster series in which there are no nodal circles. The key to their development is a grouping of the site-site total and direct correlation functions into four classes depending upon how the root points are intersected by s-bonds. They write... 

Using the diagrammatic approach, the atomic hypernetted chain (HNC) closure is obtained by neglecting all diagrams which are free of nodal circles (bridge diagrams) in the cluster expansion of the pair correlation function [6], The closure based on HNC theory is shown in Eq. (26). 

Figure 2.6 Nodal properties of standing waves. A one-dimensional oscillation (wave) constrained within a space of length L can have amplitudes (wavefunctions) of discrete wavelengths only. The open circles are the nodes where the amplitude is always zero...

Fig. 7. (a) Wave pattern generated on a circle with scaled radius kr = 1.82. The pattern is the product of a radial part, J,(kr) [the first-order Bessel function] and an angular part, cos ( >- The dashed nodal line of zero (i.e., steady state) concentration runs along diameter of the circle from < > = 90° to < > = 270°. The dotted circle outlines the circular radius. (b) Wave pattern, JAkr) cos 2dashed lines are crossed nodal lines on two perpendicular diameters, (c) Pattern generated at a scaled radius of 3.8, where the zero in the derivative of JAkr) matches the radial boundary condition. The pattern is JAkr) cos 0c >, which has no angular variation. The nodal line is concentric with the outer radius. 

Fig. 10.16. Final rotational state distributions of NO following the dissociation of C1NO through the T state. The quantum numbers n and k specify the initial vibrational and bending excitation of the ClNO(Ti) complex. The undulations for the excited bending states reflect the nodal structures of the dissociation wavefunction at the transition state. The open and the filled circles indicate different P and Q branches. The corresponding absorption spectrum is depicted in Figure 7.14. Adapted from Qian, Ogai, Iwata, and Reisler (1990).

Figure 3.10 Example of a phase diagram for a ternary system used to create a dynamic LLC system. Components Ethanol (EtOH), Acetonitrile (ACN) and Iso-octane (2,2,4-trimethylpentane TMP). I — V nodal lines. Circles compositions determined experimentally by titration (full circles) and GC (open circles). Figure taken from ref. [315]. Reprinted with permission.

Four-circle geometry. Since the crystal rotates around three orthogonal axes, this instrument geometry may seem inappropriate to measurements at very low temperatures. In fact, several good mechanical solutions have been found. Here the main constraints come from the absence of a nodal plane (or axis) for the multiaxial sample rotations, from space limitations due to the internal diameter of the x cradle, from the mechanical strength of the x cradle, which must hold the cryostat without deformation, and from the power of motors, which must be able to rotate heavy parts and to drag more or less flexible links. 

Figure 5. Nodal surfaces of a sp d2z2 hybrid orbital with Z = 1 in the momentum-space representation. The left-hand plot contains two surfaces. One is the spherical node of the imaginary part. The second more complex surface consists of two closed and flattened spheres. These are the nodal surfaces belonging to the real part of the hybrid and are aligned along the -axis. The intersection of the two types of nodes are two circles around the -axis. The right-hand plot displays a cut through the a -plane. Note that the (polar) -axis is the horizontal axis in this plot. To avoid confusion, the nodal planes of the imaginary part are not displayed in either graph.

Figure 7. Circular sections through the momentum densities displayed in Fig.(6). The plots display the value of the density along a circle of radius 1/3 in the xz-plane, as a function of the polar angle 0 = tv. The nodal points are clearly visible.

Figure 2.14. Frontier orbitals of benzene and [8)annulene dianion. The nodal planes ( ) are obtained from the polygons with 2k vertices inscribed into the perimeter, and the magnitude of the LCAO coefficients (indicated by the size of the circles) may be estimated from the location of the nodal planes.

Fig. 46. The [(U02)3(Se04)5]" sheets in the crystal structures of a- (a) and 3-Mg2[(U02)3(Se04)5](H20)i6 (b) and their nodal representations (c and d, respectively). Legend [UO7] bipyramids = black circles [Se04] tetrahedra = white circles.

Fig. 10. The [(U02)(Se04)2] sheet in the structure of (H30MCi2Ft3oN2]3[(U02)4(Se04)8] (Ft20)5 (a), its nodal representation (U and Se polyhedra are symbolized by black and white circles, respectively) (b), and organization of 1,12-dodecanediamine molecules into a micelle (c)...

Fig. 39. Overlap between d orbital neighbors in the bcc structure. For clarity second nearest neighbors have been moved in to sit in the centers of the faces of the cube. Empty circles represent atoms which lie on a a nodal plane filled circles are those which don t. The — y and xy orbitals are taken as representatives of the e and t g sets respectively. For the case of e, orbitals there is no overlap with first nearest neighbors, and no overlap of the tjg orbitals with second nearest neighbors...

For r = 1, V = y so that y is the value of the solid harmonic in the surface of the unit sphere at points defined by the coordinates 6 and surface harmonics of degree /. The associated Legendre polynomials Pf(cosd) have l — m roots. Each of them defines a nodal cone that intersects a constant sphere in a circle. These nodes, as shown in Figure 20-5, are in the surface of the sphere and not at r = 0 as assumed in the definition of atomic orbitals. Surface harmonics are obviously undefined for r = 0. The linear combinations i/ i i/i i define one real and one imaginary function directed along the X and Y Cartesian axes respectively, but these functions (denoted and ipy) are no longer eigenfunctions of L, but of or Ly instead. 

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