Bohr: radius

The Is orbital /i = is correct but not normalized. The normalized function governing the probability of finding an electron at some distance r along a fixed axis measured from the nucleus in units of the Bohr radius oq = 5.292 x 10 " m is... 

Zg is the effective charge number in the interaction of two unlike atoms, and is the Bohr radius for the hydrogen atom, 0.5292 x 10 cm. There exist a number of approximations for Z but a simple description based on a mean value is as follows. 

To understand the origins of dispersion forces, let us consider two Bohr atoms, each of which consists of an electron orbiting around a nucleus comprised of a proton, having a radius ao, often referred to as the first Bohr radius . It is obvious that a Bohr atom has no permanent dipole moment. However, the Bohr atom can be considered to have an instantaneous dipole moment given by... 

The fundamental equations of quantum chemistry are usually expressed in units designed to simplify their form by eliminating fundamental constants. The atomic unit of length is the Bohr radius ... 

Figure Al.l Radial functions for a hydrogen atom. (Note that the horizontal scale is the same in each graph but the vertical scale varies by as much as a factor of 100. The Bohr radius Oq = 52.9 pm.)...

The symbol ao is the first Bohr radius, approximately 52.9 pm, and to is the permittivity of free space.) As we will see in later chapters, Gaussian orbitals are... 

It is instructive to look at the form of the Is, 2s and 3s orbitals (Table 9.1). By convention, we use the dimensionless variable p = Zrjaa rather than r. Here 2 is the nuclear charge number and oq the first Bohr radius (approximately 52.9 pm). The quantity Z/n is usually called the orbital exponent, written These exponents have an increasing number of radial nodes, and they are orthonormal. 

The quantity a ) is called the Bohr radius when the values of the fundamental constants are inserted, we find a0 = 52.9 pm. 

What does this equation tell us For this wavefunction, the angular wavefunction Y is a constant, 1/2tti/2 , independent of the angles, which means that the wavefunction is the same in all directions. The radial wavefunction R(r) decays exponentially toward zero as r increases, which means that the electron density is highest close to the nucleus (e° =1). The Bohr radius tells us how sharply the wavefunction falls away with distance when r = a ), t i has fallen to 1/e (37%) of its value at the nucleus. 

Bohr frequency condition, 13 Bohr radius, 23 boiling, 314 boiling point, 314 alkanes, 737 anomalous, 184 boiling point prediction, 180 boiling point trends, 183 boiling-point elevation, 332 Boltzmann, L., 276 Boltzmann formula, 276 bomb calorimeter, 224... 

Fig. 5.6 Changes in the shape of the valence contribution due to geometric and electronic relaxation in [FeF4]" . Full line [FeF4] at its equilibrium geometry, dashed line [FeF4] at its equilibrium geometry. The square of the valence orbital that mainly contributes to p(0) along the Fe-F bond (distances are in units of the Bohr radius) is also drawn (from [19])...

The units we use in daily life, such as kilogram (or pound) and meter (or inch) are tailored to the human scale. In the world of quantum mechanics, however, these units would lead to inconvenient numbers. For example, the mass of the electron is 9.1095 X J0 31 kg and the radius of the first circular orbit of the hydrogen atom in Bohr s theory, the Bohr radius, is 5.2918 X 10 11 m. Atomic units, usually abbreviated as au, are introduced to eliminate the need to work with these awkward numbers, which result from the arbitrary units of our macroscopic world. The atomic unit of length is equal to the length of the Bohr radius, that is, 5.2918 X 10 n m, and is called the bohr. Thus 1 bohr = 5.2918 X 10"11 m. The atomic unit of mass is the rest mass of the electron, and the atomic unit of charge is the charge of an electron. Atomic units for these and some other quantities and their values in SI units are summarized in the accompanying table. 

The abbreviation au stands for atomic units , which is a system of units meant to simplify the equations of molecular and atomic quantum mechanics. The units of the au system are combinations of the fundamental units of mass (mass of the electron), charge (charge of the electron), and Planck s constant. By setting these three quantities equal to unity one gets simpler equations. The au system has a simple relation to the SI and Gaussian (cgs) systems of units. For example, 1 au of length = aQ (Bohr radius) = 5.29 x 10"9 cm =... 

PbS has attracted much attention due to its special direct band gap energy (0.4 eV) and a relatively large exciton Bohr radius (18 nm) and their nanoclusters have potential applications in electroluminescent devices such as light-emitting diodes. PbS nanocrystals with rod like structures with diameters of 20-60 nm and lengths of 1-2 pm have been obtained using the sonochemical method and by using PEG-6000 [66]. Addition of PEG and the time of sonication have been found to play a key role in the formation of these rods. 

Deutsch and Mark compared the classical expression with a theory developed by Bethe.37 Bethe s calculations showed that the ionization cross section for an atomic electron is approximately proportional to the mean square radius of the appropriate n,l electronic shell. Experiment had also shown a correlation between the maximum in the atomic cross section and the sum of the mean square radii of all outer electrons. This led to the replacement of the Bohr radius with the radius of the corresponding subshell the ionization cross section is now given by,... 

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