Atomic unit

Atomic unit for action (energy x time) or moment of momentum Planck s constant h divided by 2n (h = 1). 

Atomic unit for charge absolute value of charge of the electron ( e = 1). 

Atomic unit for mass rest mass of the electron (m = 1). 

Atomic unit for permittivity 4% times the permittivity of vacuum (4toq = 1). 

Since the proton and neutron are 1836 times as heavy as the electron, the atomic nuclei move quite slowly (on the femtosecond scale) compared with the electrons. It is possible to measure the motion of nuclei in fast chemical reactions 

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 

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. 

When atomic units are used, one sets e = hj2ir = m9 — 1 in quantum mechanical equations. For example — AaVa/ 7r tne becomes — V.  

The advantage of atomic units is that if all the calculations are directly expressed in such units, the results do not vary with the subsequent revision of the numerical values of the fundamental constants. 

The reader may confirm that the content of these equations is the same as that of eqns (10-3.1) to (10-3.6). 

The units we use throughout this book are called atomic units. To see how these units arise naturally let us consider the Schrodinger equation for the hydrogen atom. In SI units, we have 

The constants in front of the kinetic and potential energy operators can then be factored, provided we choose A such that 

Thus A is just the Bohr radius Gq which is the atomic unit of length called a Bohr, Finally, since 

Conversion factors for a few other units, which are not related to SI but which are necessary to read the existing literature, are as follows. One atomic unit of length equals 0.52918 Angstroms (A). One atomic unit of dipole moment (two unit charges separated by ao) equals 2.5418 Debyes (D), and one atomic unit of energy equals 27.211 electron volts (eV) or 627.51 kcal/mole. 

From now on we drop the primes and all our quantities will be in atomic units. 

Most quantum chemists report the results of their calculations using atomic units. 

First consider the cgs Gaussian system of units. The hydrogen-atom Hamiltonian in these units is (assuming infinite nuclear mass) 

The atomic unit of energy, e ao, is caUed the hartree (symbol E  

The ground-state energy of the hydrogen atom is hartree if nuclear motion is neglected. The atomic unit of length is called the bohr  

A more rigorous way to define atomic units is as follows. Starting with the H-atom electronic Schrddinger equation in SI units, we define (as in Section 4.4) the dimensionless 

The hydrogen-atom Hamiltonian operator (assuming infinite nuclear mass) in SI units is — /2m )V — /AiTE r. The system of atomic units is defined as follows. The units 

The ground-state energy of the hydrogen atom is given by (6.94) as — ( /47 8o o) Since [Eq. (6.106)] Uq = I, the numerical value of Uq (the Bohr radius) in atomic 

It is convenient to define a system of units that is more natural for working with atoms and molecules. The commonly accepted system of atomic units for some important quantities is summarized in Table 4-1. [Note the symbol h ( h-cross or h-bar ) is often used in place of hlln.] Additional data on values of physical quantities, units, and conversion factors can be found in Appendix 10. 

In terms of these units, Schrodinger s equation and its resulting eigenfunctions and eigenvalues for the hydrogenlike ion become much simpler to write down. Thus, the 

Quantity Atomic unit in cgs or other units Values of some atomic properties in atomic units (a.u.) 

Chapter 4 The Hydrogenlike Ion, Angular Momentum, and the Rigid Rotor 

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  

Coordinates can be transformed to bohrs by dividing them by Aq. Energies are measured in hartrees, defined as the Coulomb repulsion between two electrons separated by 1 bohr  

Masses are also specified in terms of electron mass units (i.e. define mj=l). 

The Born-Oppenheimer approximation is the first of several approximations used to simplify the solution of the Schradinger equation. It simplifies the general molecular problem by separating nuclear and electronic motions. This approximation is reasonable since the mass of a typical nucleus is thousands of times greater than that of an electron. The nuclei move very slowly with respect to the electrons, and the electrons react essentially instantaneously to changes in nuclear position. Thus, the electron distribution within a molecular system depends on the positions of the nuclei, and not on their velocities. Put another way, the nuclei look fixed to the electrons, and electronic motion can be described as occurring in a field of fixed nuclei. 

The full Hamiltonian for the molecular system can then be written as  

To get rid of all fundamental physical constants in our mathematical formulae we shall introduce consistently a system of atomic units (au), by putting  

The basic atomic units are obtained from the SI values of the fundamental physical constants given in Table 1.1 (Mohr and Taylor, 2003). 

The basic au of charge, length, energy and time are then expressed by e = 1.602 176 x 10 C 

When the atomic unit of energy is referred to molar quantities, we have the different SI equivalents  

The basic au for dipole, quadrupole and octupole electric moments are given as  

For all discussions of atomic structure and dynamics caused by some external interaction, the natural measure with which to compare is given by the corresponding quantity of the atom itself. Hence, the hydrogen atom is used as a standard and provides the atomic units (au). In addition, if atomic units are used in theoretical expressions, the equations look somewhat simpler. The most important atomic units needed in the present context are (for the numerical values see [CTa87])  

The ifjs are called pseudostates, and give best E2 in the form  

The Bohr radius Oo = 0.529177 x 10 m is a unit of length. This unit is called Bohr (atom unit, a.u.). 

The rest mass of the electron, which is equal to me = 9.109 534 x 10 kg, is accepted as the unit of the mass. 

In order to convert SI units into atomic units one has to put m = ti = e = 

to determine the value of distance in meters one should multiply the value expressed in Bohr by Oq- Conversely, in order to obtain the value of distance in atomic units (a.u.) one should divide the value expressed in meters by ao = 0.529177 x 10 and so on. 

Interatomic Bonding in Solids Fundamentals,Simulation,andj plications, First Edition. Valim Levitin. 

For a water molecule we have 10 electrons and 3 nuclei but the number of two-electron interactions is ri) n — l)/2 = (10)(9)/2 = 45 since each of the 10 electrons can interact with 9 other electrons but each interaction is only counted as one interaction Similarly, the repulsions between a nucleus with charge and another nucleus with charge Zm separated by distanceare only counted once since two interacting nuclei only have one interaction but those terms are very easy to compute  

Dohgane, T. Takabatake, and M. Bersohn, Reel. Trav. Chim. Pays. Bas., 1992, 111, 291-296. 

Langley, J. M. Zytkow, H. A. Simon, and G. L. Bradshaw, in The Search for Regularity Four Aspects of Scientific Discovery , eds. R. S. Michalski,J. G. Carbonell, andT. M. Mitchell, Machine Learning, II, Morgan Kaufmann, Los Altos, CA, 1986, pp. 425-470. 

Walters, Machine Learning in Chemistry , PhD Dissertation, University of Arizona, 1993. 

Struct.-Act. Correl. Predict. Tool Toxicol. [Symp Paper], 1981, 151-169. 

Pitman, R. T. Koehler, B. S. Kislin, and G. D. Anderson, INVENTON - a system to invent chemical stmetures. Abstracts of Papers of the ACS, American Chemical Society, Washington DC, 1991, p. 202, 16-ClNF. 

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MarkownikofT s rule The rule states that in the addition of hydrogen halides to an ethyl-enic double bond, the halogen attaches itself to the carbon atom united to the smaller number of hydrogen atoms. The rule may generally be relied on to predict the major product of such an addition and may be easily understood by considering the relative stabilities of the alternative carbenium ions produced by protonation of the alkene in some cases some of the alternative compound is formed. The rule usually breaks down for hydrogen bromide addition reactions if traces of peroxides are present (anti-MarkownikofT addition). 

The atomic unit (AU) of dipole moment is that of a proton and electron separated by a distance equal to the first Bohr orbit, oq. Similarly, the au of polarizability is Oq [125]. Express and o for NH3 using both the cgs/esu and SI approach. 

Figure Bl.19.7. A series of time-lapse STM topographic images at room temperature showing a 40 mn x 40 mn area of Au(l 11). The time per frame is 8 mm, and each took about 5 min to scan. The steps shown are one atomic unit in height. The second frame shows craters left after tip-sample contact, which are two and three atoms deep. During a 2 h period the small craters have filled completely with diflhismg atoms, while the large craters continue to fill. (Taken from [29], figure 1.)...

For H2, let us write down the zeroth-order electronic Hamiltonian (in atomic unit) ... 

The next question asked is whether there are any indications, from ab initio calculations, to the fact that the non-adiabatic transfonnation angles have this feature. Indeed such a study, related to the H3 system, was reported a few years ago [64]. However, it was done for circular contours with exceptionally small radii (at most a few tenths of an atomic unit). Similar studies, for circular and noncircular contours of much larger radii (sometimes up to five atomic units and more) were done for several systems showing that this feature holds for much more general situations [11,12,74]. As a result of the numerous numerical studies on this subject [11,12,64-75] the quantization of a quasi-isolated two-state non-adiabatic coupling term can be considered as established for realistic systems. 

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. 

Figure 6.25 reprinted from Chemical Physics Letters, 196, Ding H-Q, N Karasawa and W A Goddard III, T he Reduced Cell Multipole Method for Coulomb Interactions in Periodic Systems with Million-Atom Unit Cells, 6-10, 1992, with permission of Elsevier Science. 

Quantum mechanics is primarily concerned with atomic particles electrons, protons and neutrons. When the properties of such particles (e.g. mass, charge, etc.) are expressed in macroscopic units then the value must usually be multiplied or divided by several powers of 10. It is preferable to use a set of units that enables the results of a calculation to he reported as easily manageable values. One way to achieve this would be to multiply eacli number by an appropriate power of 10. However, further simplification can be achieved by recognising that it is often necessary to carry quantities such as the mass of the electron or electronic charge all the way through a calculation. These quantities are thus also incorporated into the atomic units. The atomic units of length, mass and energy are as follows ... 

Using atomic units, the Schroedinger equation for ground-state hydrogen is... 

One fine summer evening, I was returning by the last omnibus, outside as usual, through the deserted streets of the metropolis, which are at other times so full of life. I fell into a reverie and lo the atoms were gamboling before my eyes.. .. I saw how, frequently, two smaller atoms united to form a pair, how a larger one embraced two smaller ones how still larger ones kept hold of three or even four of the smaller whilst the whole kept whirling... 

Note that these equations do not contain the constants that are typically included in introductory texts, such as the vacuum permitivity constant. Theoreticians, and thus software developers, work with a system of units called atomic units. Within this unit system, many of the fundamental constants are defined as having a value of 1. Atomic units will be used throughout this book unless otherwise specified. 

The system of atomic units was developed to simplify mathematical equations by setting many fundamental constants equal to 1. This is a means for theorists to save on pencil lead and thus possible errors. It also reduces the amount of computer time necessary to perform chemical computations, which can be considerable. The third advantage is that any changes in the measured values of physical constants do not affect the theoretical results. Some theorists work entirely in atomic units, but many researchers convert the theoretical results into more familiar unit systems. Table 2.1 gives some conversion factors for atomic units. 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

APW (augmented plane wave) a band structure computation method atomic mass unit (amu) atomic unit of mass... 

Hamiltonian quantum mechanical operator for energy, hard sphere assumption that atoms are like hard billiard balls, which is implemented by having an infinite potential inside the sphere radius and zero potential outside the radius Hartree atomic unit of energy... 

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