Atoms wavefunctions

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as... 

We often refer to Heitler and London s method as the valence bond (VB) model. A comparison between the experimental and the valence bond potential energy curves shows excellent agreement at large 7 ab but poor quantitative agreement in the valence region (Table 4.3). The cause of this lies in the method itself the VB model starts from atomic wavefunctions and adds as a perturbation the fact that the electron clouds of the atoms are polarized when the molecule is formed. 

E. Clement and C. Roetti, Roothan-Hartree-Fock atomic wavefunctions,... 

Alternative methods are based on the pioneering work of Hylleraas ([1928], [1964]). In these cases orbitals do not form the starting point, not even in zero order. Instead, the troublesome inter-electronic terms appear explicitly in the expression for the atomic wavefunction. However the Hylleraas methods become mathematically very cumbersome as the number of electrons in the atom increases, and they have not been very successfully applied in atoms beyond beryllium, which has only four electrons. Interestingly, one recent survey of ab initio calculations on the beryllium atom showed that the Hylleraas method in fact produced the closest agreement with the experimentally determined ground state atomic energy (Froese-Fischer [1977]). 

The angular momenta of atoms are described by the quantum numbers L, S or J. When spin-orbit coupling is important, it is the total angular momentum J which is a constant of the system. A group of atomic wavefunctions with a common J value - akin to a term, as described in Section 3.6 - comprise (27 -i- 1) members with Mj... 

For calculational purposes, the basic quantities rM, Q and Oe(rJ ) are obtained as follows a) From accurate atomic wavefunctions, the electron density is obtained by numerical integration (b) with the electron density at any point at hand, the values are obtained by the simple reading of the distance at which p r) attains the critical value of 0.00872 electron units and (c) the central charge and the RF potential are readily obtained in terms of ty and Og(/>) via Eqs (11) and (15), respectively. 

Clementi, E. and Roetti, C., Roothaan-Hartree-Fock Atomic Wavefunctions, Atomic Data and Nuclear Data Tables 14, 177 (1974). 

The atomic wavefunctions are fixjm Ref. 212. Additional results are reported in Refs. 27 and 213. 

A widely used approximate method of describing atomic states is the Slater atomic wavefunctions (Zener, 1930 Slater, 1930). In this section, we show that regarding STM, the Slater wavefunction is a convenient tool for describing localized atomic states at surfaces. 

In treating complex atoms and molecules, these hydrogenlike wavefunctions are still too complicated in most applications. In treating low-energy (up to a few eV) problems, the behavior of the wavefunctions near the nuclei of the atoms is not important. Therefore, in the hydrogen wavefunctions, only the term with the highest power of r is significant. Based on this observation, Zener (1930) and Slater (1930) proposed a simplified form of the atomic wavefunctions ... 

The spirit of the Slater atomic wavefunction is to use the term with the highest power r" to represent the entire algebraic factor in the hydrogenlike wavefunctions. This approach is particularly suitable for treating STM-related problems. In the processes pertinent to STM, only the values of atomic wavefunctions a few Angstroms away from the nucleus of the atom are relevant, not the values near the core. Within the same spirit, we can derive all the Slater functions from a single function... 

By taking derivatives with respect to z, a factor xir = cos 6 is generated. Within the same approximation of the Slater atomic wavefunctions, the differentiation acts on the exponential factor only. Therefore,... 

DAS model 16 resolved in real space 10 Slater atomic wavefunctions 149 Sommerfeld metal 93 Space groups 357 Spherical harmonics 76, 344—345 real form 344... 

Plot the radial portions of the 4s, 4p, 4d, and 4f hydrogen like atomic wavefunctions. 

In the Koopmans theorem Umit the photoemission of one-electron from an atom or a core in a solid is given by a single Une, positioned at the eigenvalue of the electron in the initial state. The intensity of this line depends on the cross-section for the event, which is determined by the one-electron atomic wavefunctions Wi ( j m)(-Eb) and Pfln(nM, m )(Ekin) (where the atomic quantum numbers are indicated as well as the eigenvalues En,i,m = Eb and E dn of the initial and final state) (the overlap integral of (13)... 

There is no hard boundary between the overlap dependent and independent regions. One can only ask at which point the overlap dependent energies become comparable to the overlap independent. If we take Hj as an example, the separation of the lowest singlet and triplet state depends primarily on overlap (in any model based upon atomic wavefunctions) whereas the absolute energies contain overlap independent terms. Table 2 shows these as a function of interatomic distance 104, IOS with the Coulomb (overlap independent) and exchange (overlap dependent) terms being given by... 

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

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