Jacob’s Ladder

Symbolism is the language of the subconscious mind, used in dream and vision. We have access to the entire store of symbols held in the collective unconscious. In Qabalah, this is attributed to the astral realm of Yetzirah on Jacob s Ladder, and to the Sephira of Yesod on the simple Tree. It is called the treasure house of images. As conscious beings, we... 

Figure 10.2 Illustration of Perdew s classification of DFT functionals using Jacob s ladder, reaching rung by rung between Earth and Heaven. The physical ingredients included in the functionals from each rung are summarized on the right.

The third rung of Jacob s ladder is defined by meta-GGA functionals, which include information from n r), Vn(r), and V2n(r). In practice, the kinetic energy density of the Kohn-Sham orbitals,... 

The simplest approximation to Fxc[p(r)L the bottom rung of the DFT Jacob s ladder, results from the local density approximation, LDA. In mathematics a local property of a function at a point on the surface (line, or two-dimensional surface, or hypersurface) that is defined by the function is a property that depends on the behavior of the function only in the immediate vicinity of the point [49]. Immediate vicinity can be taken to mean the region within an infinitesimal distance beyond the point. Consider the derivative at some point P on the line defined by plotting y = fix) against x. This property, the derivative or gradient, is the limit... 

In the previous sections, the following nonempirical functionals were mentioned LDA, PBE, and TPSS. These functionals represent the lowest three rungs on the Jacob s ladder LDA, GGA, and meta-GGA. At the GGA-and meta-GGA levels, the approximate exchange-correlation functionals were obtained, not directly, but through consttuction of approximations to the exchange-correlation hole (nxc(r, r ))... 

The Exchange-Correlation Functionals Climbing Jacob s Ladder... 

Perdew has described a hierarchy of approximate treatments of the exchange-correlation term as Jacob s Ladder. The ladder is grounded in HF theory here on earth, and reaches to heaven, where the exact functional is found. Along the way are five rungs, each defining a set of assumptions made in creating an exchange-correlation expression. 

In order to make improvements over the LSDA, one has to assume that the density is not uniform. The approach that has been taken is to develop functionals that are dependent on not only the electron density but also derivatives of the density. This constitutes the generalized gradient approximation (GGA) and is the second rung on Jacob s Ladder. The third rung, meta-GGA functional, includes a dependence on the Laplacian of the density (V p) or on the orbital kinetic energy density (t). The fourth row, the hyper-GGA or hybrid functionals, includes a dependence on exact (HF) exchange. Finally, the fifth row incorporates the unoccupied Kohn-Sham orbitals. This is most widely accomplished within the so-called double hybrid functionals. 

Perdew, J. P. Schmidt, K. Jacob s ladder of density functional approximations for the exchange-correlation energy, In Density Functional Theory and its Application to Materials Doren, V. V, Alsenoy, C. V, Geerlings, P, Eds. AlP-Press Melville,... 

Synonyms Convallaria Convallaria majalis Con-vall-lily Jacob s Ladder Ladder-to-Heaven Lilje-konvall Lily Constancy Lily convalle Male Lily May Lily Our Lady s Tears... 

The colorful rainbow, that appears above the new Earth, is a sign that the flood has reached its end, and there is peace between Water and Fire. It is the mystical Jacob s Ladder, which stands between Heaven and Earth, and on which God s Angels ascend and descend. 

The gliding arc has been known for more than one hundred years in the form of Jacob s ladder and was first used for producing nitrogen-based fertilizers by Naville and Guye (1904). Recent contributions to the development of gliding arcs have been made by A. 

As we saw in Chapter 17, an initiate must pass through the luminous darkness that surrounds the Pole to achieve a reunion with his own dimension of light. Then he must look vertically upwards from the terrestrial toward the celestial pole. The stairway to heaven is a Jacob s ladder. It leads to the constellation of the Bear, Ursus Major. This same stairway to heaven existed in Dilmun for the Sumerians and Pharaohs of Egypt. 

At the fifth level of the Jacob s ladder classification, the full information of the KS orbitals is employed, i.e. not only the occupied but also the virtual orbitals are included. The formalism here becomes similar to those used in the random phase approximation (Section 10.9), but very little work has appeared on such methods. Inclusion of the virtual orbitals is expected to significantly improve on, for example, dispersion (such as van der Waals) interactions, which is a significant problem for almost all current functionals. 

Table 6.1 shows an overview of commonly used functionals given by their acronym, and placed in the Jacob s ladder classification. One may furthermore differentiate the functionals based on their use (or lack) of experimental data for assigning values to the parameters in the functional forms. The non-empirical ones such as the PW86,... 

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