Density correspondence principle

The atom-centered models do not account explicitly for the two-center density terms in Eq. (3.7). This is less of a limitation than might be expected, because the density in the bonds projects quite efficiently in the atomic functions, provided they are sufficiently diffuse. While the two-center density can readily be included in the calculation of a molecular scattering factor based on a theoretical density, simultaneous least-squares adjustment of one- and two-center population parameters leads to large correlations (Jones et al. 1972). It is, in principle, possible to reduce such correlations by introducing quantum-mechanical constraints, such as the requirement that the electron density corresponds to an antisymmetrized wave function (Massa and Clinton 1972, Frishberg and Massa 1981, Massa et al. 1985). No practical method for this purpose has been developed at this time. 

A common reference density, first used by Roux and Daudel (1955), is the superposition of spherical ground-state atoms, centered at the nuclear positions. It is referred to as the promolecule density, or simply the promolecule, as it represents the ensemble of randomly oriented, independent atoms prior to interatomic bonding. It is a hypothetical entity that violates the Pauli exclusion principle. Nevertheless, the promolecule is electrostatically binding if only the electrostatic interactions would exist, the promolecule would be stable (Hirshfeld and Rzotkiewicz 1974). The difference density calculated with the promolecule reference state is commonly called the deformation density, or the standard deformation density. It is the difference between the total density and the density corresponding to the sum of the spherical ground-state atoms located at the positions R... 

According to the correspondence principle the classical expression for the electron density p(r,t) can be converted to the quantum mechanical description by taking into account that the particle density is calculated by integration of the product of the iV-electron wave function 4 1 and its complex conjugate 4. We introduce the charge-weighted density by multiplication of the electron density with the electron charge,... 

The wave functions for a particle in a box illustrate another important principle of quantum mechanics the correspondence principle. We have already stated earlier (and will often repeat) that all successful physical theories must reproduce the explanations and predictions of the theories that preceded them on the length and mass scales for which they were developed. Figure 4.25 shows the probability density for the n = 5, 10, and 20 states of the particle in a box. Notice how the probability becomes essentially uniform across the box, and at m = 20 there is little evidence of quantization. The correspondence principle requires that the results of quantum mechanics reduce to those of classical mechanics for large values of the quantum numbers, in this case, n. 

The problem of the real existence of the five above states of the carbon-metal bond in any specific situation will not be discussed because it is considered comprehensively in Refs. [55-57], It should only be noted that in the case of the carbon-lithium bond all five bond states appear to exist, whereas other alkali metals cannot form the slightly polar carbon-metal bond. We mean by the slightly polar bond the bond in which the electron density distribution may be assumed to be virtually symmetric (pseudosymmetric) from the viewpoint of the correspondence principle. 

One conclusion that can be reached from the early work on effective potentials [1,21-23], the work of Cao and Voth [3-8], as well as the centroid density-based formulation of quantum transition-state theory [42-44,49] is that the path centroid is a particularly useful variable in statistical mechanics about which to develop approximate, but quite accurate, quantum mechanical expressions and to probe the quantum-classical correspondence principle. It is in this spirit that a general centroid density-based formulation of quantum Boltzmann statistical mechanics is presented in the present section. This topic is the subject of Paper I, and the emphasis in this section is on analytic theory as opposed to computational approaches (cf. Sections III and IV). 

FIGURE 3.8 The quantum harmonic oscillator eigen-function probabilities (density) representation (thick continuous curves) for ground state ( = 0), and few excited vibronic states ( = 2, 5, and 10) for the working case of HI molecule (respecting the coordinated centered on its mass center) the classical potential is as well illustrated (by the dashed curve in each instant) for facihtating the correspondence principle discussion. 

In Section 24.1.3 we have discussed among others the time-temperature correspondence principle. An example of application of that principle is shown in Fig. 24.10. The results pertain to high density polyethylene (HOPE) subjected to different levels of predrawing [58]. The draw ratio is defined as... 

In order to understand the relation of the 4-current to the spin density, it is important to realize that the definition of the current density (naturally) involves a velocity operator, which is in close analogy to classical mechanics (correspondence principle), as we have seen before. As the velocity operator in Dirac s theory contains Dirac matrices a which are composed of Pauli spin matrices cr, we understand that the current density j carries the spin information. 

The quantity qt is the charge on nucleus i, and /Oe (t /) is the quantum mechanical electron probability density corresponding to the electronic state Z. Note that the right side of Eq. (14) is the force on nucleus i, as it would be calculated from classical electrostatics if /oe(t Z) was known. Thus, this equation justifies the description of (short-ranged) intra-molecular interactions via empirical force fields. We remark that Eq. (14), at least in principle, allows quantum MD calculations. For fixed positions of the nuclei, /oe(t Z) may be computed solving Schrodinger s equation numerically. Subsequently, the nuclei are displaced according to Eqs. (1) and (14). However, this procedure is prohibitively slow, and in practice other methods are used. 

The calculation of UV/vis spectra, or any other form of electronic spectra, requires the robust calculation of electronic excited states. The absorption process is a vertical transition, i.e. the electronic transition happens on a much faster timescale than that of nuclear motion (i.e. Bom-Oppenheimer dynamics, more correctly referred to as the Franck-Condon principle in the context of electronic spectroscopy). The excited state, therefore, maintains the initial ground-state geometry, with a modified electron density corresponding to the excited state. To model the corresponding emission processes, i.e. fluorescence or phosphorescence, it is necessary to re-optimize the excited-state nuclear geometry, as emission in condensed phases generally happens from the lowest vibrational level of the emitting excited state. This is Kasha s Rule. 

The use of the correspondence principle to define a quantum mechanical analog of the stress tensor in classical mechanics goes back to Schrodinger and Pauli [33, 34]. The interest of electronic structure theorists in the stress tensor has been episodic, starting with the rise of computational density functional theory in the... 

Liquid densities can be calculated according two types of methods, both based on the principle of corresponding states. 

Calculation of thermophysical properties of gases relies on the principle of corresponding states. Viscosity and conductivity are expressed as the sum of the ideal gas property and a function of the reduced density ... 

For each configuration of the nuclei, minimization of tlie total energy with respect to the electron density yields the instantaneous value of a potential energy fiinction V(/ ), and the corresponding forces on the nuclei. In principle,... 

Since the coiTelation between opposite spins has both intra- and inter-orbital contributions, it will be larger than the correlation between electrons having the same spin. The Pauli principle (or equivalently the antisymmetry of the wave function) has the consequence that there is no intraorbital conelation from electron pairs with the same spin. The opposite spin correlation is sometimes called the Coulomb correlation, while the same spin correlation is called the Fermi correlation, i.e. the Coulomb correlation is the largest contribution. Another way of looking at electron correlation is in terms of the electron density. In the immediate vicinity of an electron, here is a reduced probability of finding another electron. For electrons of opposite spin, this is often referred to as the Coulomb hole, the corresponding phenomenon for electrons of the same spin is the Fermi hole. 

Assume now that two different external potentials (which may be from nuclei), Vext and Vgjjj, result in the same electron density, p. Two different potentials imply that the two Hamilton operators are different, H and H, and the corresponding lowest energy wave functions are different, and Taking as an approximate wave function for H and using the variational principle yields... 

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