Rydberg atomic units

An atomic unit of length used in quantum mechanical calculations of electronic wavefunctions. It is symbolized by o and is equivalent to the Bohr radius, the radius of the smallest orbit of the least energetic electron in a Bohr hydrogen atom. The bohr is equal to where a is the fine-structure constant, n is the ratio of the circumference of a circle to its diameter, and is the Rydberg constant. The parameter a includes h, as well as the electron s rest mass and elementary charge, and the permittivity of a vacuum. One bohr equals 5.29177249 x 10 meter (or, about 0.529 angstroms). 

Figure 9.1. Energy level diagram for hydrogen molecule, H2, and separated atoms H R = 00) and He R = 0). R = the Rydberg constant = 13.6057 eV = 0.5 a.u. (atomic unit of energy). Value from ionization potential of He (Is 2p P). Value from ionization potential of H2. The experimental ionization potentials are quite precise but for systems containing more than one electron their interpretation in terms of orbital energies is an approximation.

In this review, atomic units will be used throughout unless otherwise noted. The most relevant atomic units for this review are the Hartree imit for energy and the Bohr unit for length. One Hartree is about 27.211 electron volts and equals 2 Rydbergs one Bohr is about 0.52918 Angstroms. More details can be foimd in Ref. [58], p. 41—43,orRef [59], p. xiv—xv. 

This is just twice the ionization potential of the hydrogen atom if the re duced mass of the electron is replaced by the rest mass. One atomic unit of energy is equivalent to twice the Rydberg constant for infinite mass. 

Another interesting example is the chaotic autoionization of molecular Rydberg states caused by the interaction of the electron with the degrees of freedom of the core. We consider the model in which the core consists of a positive Coulomb charge plus a rotating dipole that lies in the same plane of the electron orbit (m = l). The Hamiltonian reads (atomic units)... 

In developing Rydberg atom wavefunctions we begin with the Schroedinger equation for the H atom, which, in atomic units, may be written as... 

Eqs. (5.1)-(5.3) present black body radiation in a familiar form. Both to conform to the general use of atomic units in this book and to simplify the calculation of the Rydberg atoms response to the radiation we reexpress Eqs. (5.1)-(5.3) in atomic units.5... 

Fig. 19.9 Plot of scaled total decay rates n3r of Ba 6pmn( J = l + 1 autoionizing states in atomic units vs (. For ( = 0-4 the measured rates (O) shown are the average rates from many n values. The data for the rates for > 4 are for n = 12. The solid line is a simple theoretical calculation based on the dipole scattering of a hydrogenic Rydberg electron from the 6p core electron. Note that the core penetration of the lower states reduces the actual rate from the one calculated using the dipole scattering model. The constant total decay rate for > 8 is the spontaneous decay rate of the Ba+ 6p state (from ref. 39).

Laser spectroscopy, at the moment, is the method par excellanoe to measure R. Measurement of the Rydberg constant R is a simple matter. One measures the wavelength or frequency (the velocity of light is defined to be 299 792 1)58 m/sec) in a system, such as hydrogen, where theoretical calculations are expected to be accurate to within the experimental error. One then compares this measurement (in Hz or cm-1) with the theoretical calculation (in atomic units), thereby finding the atomic unit in Hz or cm-1. Half of the atomic unit is the Rydberg... 

Fig XXI-1.— Energies of electrons in the copper atom, in Rydberg units, us a function of principal quantum number n. Eneigies are shown on a logai ithmie scale. The energies in the hydrogen atom are shown for comparison. 

Figure 1. Orbital energies (in Rydberg atomic units) for the high spin form of the cubane three-iron cluster. Solid lines show spin-up orbital energies, dashed lines show spin-down. Numbers beside each line give the percent iron d-character in that orbital. The two orbitals marked with an asterisk (of a and e symmetry) are the potential locations for the final electron. The plot corresponds to the state with the final electron in the a orbital marked with an asterisk.

Table 5.3 gives a semi-empirical value of the effective Rydberg R ood for EM donors in 3C-SiC. It is the ratio of the experimental 3p i — 2p i spacing of the Nc spectrum obtained from Table 6.10 to the same spacing in atomic units, obtained by a linear interpolation of the calculated energy levels of Table 5.2 for 71/3 = 0.7181. This value of R ood (34.85 meV) is used to calculate the energies of the other donor levels by the same interpolation method. The first two rows of Table 6.10 gives the experimental positions of the lines attributed to Nc and to the EMD centre in 3C-SiC by Moore et al. [170]. The calculated... 

In the atomic units (a.u.) system, the permittivity of vacuum is dimensionless and set equal to (4ji) 1, while ao, e2, me, and ft are set equal to unity. The atomic unit of energy, the Hartree, is equal to two times the Rydberg... 

There are several conventions for the energy scale, according to context. The quantity Roo (the Rydberg) sometimes serves in theoretical work as a unit of energy, but, more usually, the Hartree (equal to 2 x the Rydberg) is preferred and is termed the atomic unit of energy.4 In atomic units, h = e = m = 47T o = 1, and equation (2.2) assumes the particularly simple form... 

One problem in which dynamics comes to the fore is the ionisation of a Rydberg atom exposed to an unspecified number of oscillations of a microwave beam. It turns out that this problem can be treated semiclas-sically and in one dimension since the atom is in a high Rydberg state, the microwave field can be a strong perturbation, and the time dependence of the Hamiltonian becomes important. In its simplest form (in atomic units)... 

In the second line EXCHF is the Slater Xa factor, but since IXCH = 0 this is not used. The next four numbers are the energy increments in Rydbergs used in the first two iterations to evaluate the energy derivatives. As mentioned in the paragraph about Step 3 in Sect.9.6, later iterations use a fraction of the relevant bandwidth. Further, RKEY = 1 R1 indicates that the entries in Line 4 are the radii in atomic units rather than a percentage of the radius used in the original band calculation. 

Just to make things more difficult (so it seems), yet another choice of atomic units is preferred in the theoretical solid-state physics community By dropping the 1/2 in front of the kinetic energy operator V, the energy directly becomes —13.606 eV, and the new unit is called one Rydberg. Thus, one Hartree equals two Rydbergs, and there are two types of atomic unit. 

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