First Bohr radius

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 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. 

Thus, for the hydrogen atom (Z = 1) the most probable distance of the electron from the nucleus is equal to the radius of the first Bohr orbit. 

The effect of the spin-orbit interaction term on the total energy is easily shown to be small. The angular momenta L and S are each on the order of h and the distance r is of the order of the radius ao of the first Bohr orbit. If we also neglect the small difference between the electronic mass We and the reduced mass the spin-orbit energy is of the order of... 

It is noticed that a = npe2/h2eo = 1/ri is inversely proportional to the first Bohr radius. The corresponding energy... 

Electronic and nuclear energy in H2. a. Values for non-interacling electrons. 6, Coulomb energy of nuclear repulsion, c, Approximate electronic energy curve for interacting electrons. Units ordinates, 1 = Rydberg constant, abscissas, 1 = radius of first Bohr orbit in hydrogen atom. 

They are given by the dashed line curves in Fig. 2.12. A conceptually useful quantity is the probability of finding the electron at some distance, r, from the nucleus (in any direction) which is determined by the radial probability density, P ,(r) = r2R (r). We see from the full line curves in Fig. 2.12 that there is a maximum probability of locating the electron at the first Bohr radius, aly for the Is state and at the second Bohr radius, a2, for the 2p state. 

The energy and length scales that are appropriate at the atomic level are those set by the ionization potential and first Bohr radius of the hydrogen atom. In SI units the energy and radius of the nth Bohr stationary orbit are given by... 

The atomic unit of length is the radius of the first Bohr orbit in the hydrogen atom when the reduced mass of the electron is replaoed by the rest mass tne. Thus the atomic unit of length is... 

If the relativistic effects are sufficiently large and therefore cannot be accounted for as corrections, then as a rule one has to utilize relativistic wave functions and the relativistic Hamiltonian, usually in the form of the so-called relativistic Breit operator. In the case of an N-electron atom the latter may be written as follows (in atomic units, in which the absolute value of electron charge e, its mass m and Planck constant h are equal to one, whereas the unit of length is equal to the radius of the first Bohr orbit of the hydrogen atom) ... 

Show by substitution in the formula given in the text ( Interaction of Light with Matter ) that ao, the radius of the first Bohr orbit for hydrogen, is 5.29 x 10-11 m. 

The unit of length is ao, the first Bohr radius of atomic hydrogen. The Hartree unit of energy is e2/ao, approximately 27.212 electron volts. 

Length Radius of first Bohr orbit (1 bohr = a0) 107(/i/27iec)2me 1 5.2917725 x 10-11m... 

Figure 7 A schematic representation of dopant atoms in a Si lattice (a) P-doped Si lattice, and (b) B-doped Si lattice. The dopant atoms are shown as replacing a Si atom in the crystal lattice. The circles represent schematic Bohr radii of the carrier (i.e. an electron (—) or a hole (+)) around the dopant atom. Note that these orbits are not drawn to scale in reality, the first Bohr radius of these carriers is about 12 A, and a single carrier is spread over about 10 Si atoms...

The sizes and shapes of the hydrogen atom orbitals are important in chemistry because they provide the foundations for the quantum description of chemical bonding and the molecular shapes to which it leads. Sizes and shapes of the orbitals are revealed by graphical analysis of the wave functions, of which the first few are given in Table 5.2. Note that the radial functions are written in terms of the dimensionless variable a, which is the ratio of Zr to ao- For Z = 1, a- = 1 at the radius of the first Bohr orbit of the hydrogen atom. 

FIGURE 5.10 Dependence of radiai probabiiity densities on distance from the nucieus for one-eiectron orbitais with n = 1, 2, 3. The smaii arrow be-iow each curve iocates the vaiue of r f for that orbitai. The distance axis is expressed in the same dimensioniess variabie introduced in Tabie 5.1. The vaiue 1 on this axis is the first Bohr radius for the hydrogen atom. Because the radiai probabiiity density has dimensions (iength) the caicuiated vaiues of are divided by... 

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