Atomic units, definition

Besides the expressions for a surface derived from the van der Waals surface (see also the CPK model in Section 2.11.2.4), another model has been established to generate molecular surfaces. It is based on the molecular distribution of electronic density. The definition of a Limiting value of the electronic density, the so-called isovalue, results in a boundary layer (isoplane) [187]. Each point on this surface has an identical electronic density value. A typical standard value is about 0.002 au (atomic unit) to represent electronic density surfaces. 

All equations given in this text appear in a very compact form, without any fundamental physical constants. We achieve this by employing the so-called system of atomic units, which is particularly adapted for working with atoms and molecules. In this system, physical quantities are expressed as multiples of fundamental constants and, if necessary, as combinations of such constants. The mass of an electron, me, the modulus of its charge, lei, Planck s constant h divided by lit, h, and 4jt 0, the permittivity of the vacuum, are all set to unity. Mass, charge, action etc. are then expressed as multiples of these constants, which can therefore be dropped from all equations. The definitions of atomic units used in this book and their relations to the corresponding SI units are summarized in Table 1-1. 

For quantitative considerations it is convenient to use atomic units (a.u.), in which h = eo = me = 1 (me is the electronic mass) by definition. They are based on the electrostatic system of units so Coulomb s law for the potential of a point charge is = q/r. Conversion factors to SI units are given in Appendix B here we note that 1 a.u. of length is 0.529 A, and 1 a.u. of energy, also called a hartree, is 27.211 eV. Practically all publications on jellium use atomic units, since they avoid cluttering equations with constants, and simplify calculations. This more than compensates for the labor of changing back and forth between two systems of units. 

The definitions of the first and second order magnetic perturbation operators are given helow. In the nonrelativistic formalism these operators are two-component operators, in the Kutzelnigg formalism all operators are to he multiplied hy the four-component matrix. All operators are given in the atomic unit system and we do not apply QED corrections so that the free electron g-factor is precisely equal to 2. 

Figure 7.4 Definition of the phase shift A as introduced by a potential. The solution of the radial function RKAr) of a wave with energy e = k2/2 (in atomic units) and with ( = 0 is shown for two situations under the influence of a repulsive potential V(r) as indicated by the shaded region (top), and for vanishing potential (bottom). In the first case one has RK((r) = FK0(r), and in the second case the radial function is equal to the spherical Bessel function, i.e., RKAr) = j0(fcr). Asymptotically, both solutions, FK0(r) and j0(Kr), differ only by a constant distance A in the r coordinate which is related to the phase shift A( as indicated. From The picture book of quantum mechanics, S. Brandt and H. D. Dahmen, 1st edition, 1985, John Wiley Sons Inc., NY. 1985 John Wiley Sons Inc.

For simplicity, the effective mass m was considered as the isotropic one. Note that the effective mass m in atomic units derives from the relation between the genuine effective mass and the electron mass. The electron contribution in Green s function (21), taking into account definitions (22) and (24), is equal to... 

Table 2.12 BE of furan, thiophene and selenophene in cyclohexane. Comparison of theory (PCM/DFT/B3LYP both for the geometry optimization and for the properties, aug-cc-pVTZ basis set) and experiment. See text for definitions and ref. [32] for further details. In particular, i/ + (S/oijC ) indicates the numerator of the right-hand side of Equation (2.227). Atomic units. A =632.8 nm. CM stands for the centre of nuclear masses, chosen as reference origin in the calculations...

Table 10.2 presents the kinetic information for the main reactions, in which the frequency factors have been calculated from turnover-frequency (TOF) data [8, 9]. This term, borrowed from enzymatic catalysis, quantifies the specific activity of a catalytic center. By definition, TOF gives the number of molecular reactions or catalytic cycles occurring at a center per unit of time. For a heterogeneous catalyst the number of active centers can be found by means of sorption methods. Let us consider that the active sites are due to a metal atom. By definition [15] we have ... 

Atoms and even the PO4 unit in a phosphate crystal may be considered as spherical balls and stacked in various configurations. The PO4 unit is held together in solids by five types of bonds. These bonds, which hold the atoms at definite distances from each other, are formed by the electronic configuration of the atoms. The equilibrium distance between... 

Thus the mole is defined such that a sample of a natural element with a mass equal to the element s atomic mass expressed in grams contains 1 mole of atoms. This definition also fixes the relationship between the atomic mass unit and the gram. Since 6.022 X 1023 atoms of carbon (each with a mass of 12 amu) have a mass of 12 grams, then... 

The experiment measures the rate of transition into a solid angle dSl subtended by the detector at scattering angles 9, energy channels i, 0 are defined by energy resolution. In atomic units the relative velocity V is ko. We use the notation dai 9,4>) for the differential cross section in this experiment. The definition (6.41) becomes... 

The convention used by Levine and Bethea to define the response functions omits the Taylor series factors in the series for the induced dipoles but includes a factor of (3/2) implicitly in the definition of the macroscopic quantity. Their ft is equivalent to jl,. Hence to relate their results to the more usual conventions, the /i-value must be multiplied by 4 x (3/2) x (3/5) = 18/5 and the y value by 4 x (3/2) = 6. Finally a factor (0.30/0.335) must be applied to allow for the change in the quartz standard. Carrying out these operations and converting to atomic units gives the values in Table 10. 

In the context of attempts to use only units based on the SI system an alternative definition of atomic units was proposed. One starts from the SI system (with 4 basic units), and measures then mass, electric charge, action and quantities of the dimension of the dielectric constant of the vacuum in units of m, e, h, and 47reo respectively. On this way one arrives at the same Hamiltonian in atomic units, as following Hartree - as long as no magnetic quantities are involved. We must therefore reconsider the system of units, when we come to electrons in the presence of magnetic fields (section 2,10). 

When we do not specify V and do not express p as —ihV, only the constants m and c are explicitly present, and we keep m in the expansion in powers of c. If we specify the attraction potential of a nucleus as —Zjr, (for either definition of atomic units) we automatically set also m... 

Dalton s hypothesis of the existence of atoms as definite quantities did not, however, meet with general acceptance. Davy, Wollaston, and others considered the quantities in which D ton had found thcr elements to unite with each other, as mere proportional numbern or equivalentSy as they expressed it, nor is it probable that Dalton s views would liave received any farther recognition until such time as they might have been exhxuned from some mxisty tome, had their publication not been closely followed by that of the results of the labors of Humboldt and of Gay Lusaac, concern ing the volumea in which gases unite with each other. 

Rearrangements similar to this have been shown definitely to take place in the case of other compounds. No simple alcohol is known in which a hydroxyl group is in combination with a carbon atom united to another carbon atom by a double bond. 

E = valence electron number per formula unit N = average bond number per semi-metal atom in a chosen defect-free reference structure N = additional electrons to be introduced upon formation of one defect in the reference structure, e.g. a defect in the diamond structure type localizes four electrons to form four lone electron pairs in the neighborhood of the defect Ec = electrons located in electron-poor clusters which, for example, may be determined according to Wade-Mingos rules or according to the 18e rule Ec = electron number in electron-poor cluster(s) composed of semi-metal atoms Ec = electron number in electron-poor cluster(s) composed of metal atoms). By definition, Zintl phases do not contain metal-centered occupied electronic states for these cases (1) may be rewritten as ... 

Atomic units have been used, and Fi is the integro-differential operator of the Hartree-Fock equations that determines the radial factors P i. The usual way of making stationary an energy functional that includes the orthonormality constraints of these radial factors introduces undetermined multipliers e(nl). As the definition of Fi introduces the same potential for all Pni, there is no need for off-diagonal undetermined multipliers, and the variation of the energy functional with the constraints yields the equation... 

---