Wave functions atomic

The characteristic structure of the actinide spectra is mainly determined by the special properties of the 5 f wave functions these, in turn, arise from the special nature of the 5 f effective potential-energy function Ueff (5f). Uetf describes the effect on the 5 f electrons of the attraction by the nucleus as well as of all the interactions with other electrons. It contains also a centrifugal term, 1(1 -I- l)/2 r, of particular importance for the properties of 5f wave functions, the effect of which will be illustrated later in more detail. 

Spatial extension, as expressed by the expectation value (r), is roughly comparable for 4 f and 5 f wave functions (Figs. 7 and 8). However, the many-electron wave functions resulting from the solution of the relativistic Dirac equation may also be used to calculate a number of physically interesting quantities, i.e. expectation values of observable 

A very relevant representation of atomic 5 f wave functions is given by Freeman and Koelling  

---

We use s, p, and d partial waves, 16 energy points on a semi circular contour, 135 special k-points in the l/12th section of the 2D Brillouin zone and 13 plane waves for the inter-layer scattering. The atomic wave functions were determined from the scalar relativistic Schrodinger equation, as described by D. D. Koelling and B. N. Harmon in J. Phys. C 10, 3107 (1977). 

D. R. Hartree, Reports on Progress in Physics 11, 113 (1948) this survey is brought up to date in D. R. Hartree, The Calculation of Atomic Structures Wiley and Sons, New York, and Chapman and Hall, London, 1957. See also R. S. Knox, Bibliography of Atomic Wave Functions/ in Solid State Physics (Seitz and Turnbull, eds.), Academic Press, New York, 1957, Yol. 4, p. 413. 

Chen, Tien Chi, J. Chem. Phys. 23, 2200, "Expansion of electronic wave functions of molecules in terms of united-atom wave functions."... 

Heitler-London wave function, 15-16 Helium atom, wave function for, 3 Heterolytic bond cleavage, 46, 51, 47,53 Histidine, structure of, 110 Huckel approximation, 8,9,10,13 Hydrocarbons, force field parameters for, 112... 

Fig. 42—2.—Energy curves for the hydrogen molecule-ion (in units e2/2ao), calculated for undistorted hydrogen atom wave functions.

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as... 

In this equation, f is the molecular wave function, is an atomic wave function, and a is a weighting coefficient that gives the relative weight in the "mix" of the atomic wave functions. The summation is... 

Integrals of this type are known as overlap integrals, and in a general way, they represent effectiveness with which the orbitals overlap in a region of space. If the subscripts are identical, orbitals on the same atom are indicated, and if the atomic wave functions are normalized, the value of such an integral is 1. As a result, we can write... 

The first and second integrals on the right-hand side of this equation evaluate to 1 because the atomic wave functions are assumed to be normalized. Therefore, the right-hand side of the equation reduces to... 

Although we have dealt with a diatomic molecule consisting of two hydrogen atoms, the procedure is exactly the same if the molecule is Li2, except that the atomic wave functions are 2s wave functions and the energies involved are those appropriate to lithium atoms. The VSIP for lithium is only 513 kj mol-1 rather than 1312kj mol-1 as it is for hydrogen. 

Representing the pz orbitals on atoms 1 and 2 by z, andz2, the combinations of atomic wave functions... 

The two bonding 7r orbitals represented by these wave functions are degenerate. The wave functions for the antibonding states are identical in form except that negative signs are used in the combination of atomic wave functions and in the normalization constants. 

Atoms do not all have the same ability to attract electrons. When two different types of atoms form a covalent bond by sharing a pair of electrons, the shared pair of electrons will spend more time in the vicinity of the atom that has the greater ability to attract them. In other words, the electron pair is shared, but it is not shared equally. The ability of an atom in a molecule to attract electrons to it is expressed as the electronegativity of the atom. Earlier, for a homonuclear diatomic molecule we wrote the combination of two atomic wave functions as... 

One of the simplest approaches to comprehensive molecular orbital calculations is the extended Hiickel method. This method was developed by Roald Hoffman in the 1960s, and it was applied to hydrocarbon molecules. From the discussion presented in Chapters 2 and 3, we know that one of the first things that has to be done is to choose the atomic wave functions that will be used in the calculations. One of the most widely used types of wave functions is that known as the Slater wave functions (see Section 2.4). In the extended Hiickel method, the molecular wave functions are approximated as... 

Because of interelectronic effects this Hamiltonian is not separable. Only when these effects are ignored may the total probability density ip ip be assumed to be a product of one-electron probability densities and the wave function a product of hydrogenic atomic wave functions... 

Herman and Skillman [79] used an HFS algorithm to calculate radial atomic wave functions and energy eigenvalues for all atoms, tabulating all results and the computer software at the same time. They treated all single electronic... 

Since the nuclei are identical Haa = Hbb and since H is hermitian Hab = Hba. For normalized Is atomic wave functions the integrals Saa = Sbb = 1. The secular equation therefore reduces to... 

Although the calculated molecular parameters De = 3.15 eV, re = 1.64 a0 do not compare well with experiment the simplicity of the method is the more important consideration. Various workers have, for instance, succeeded to improve on the HL result by modifying the simple Is hydrogenic functions in various ways, and to approach the best results obtained by variational methods of the James and Coolidge type. It can therefore be concluded that the method has the correct symmetry to reproduce the experimental results if atomic wave functions of the correct form and symmetry are used. The most important consideration will be the effect of the environment on free-atom wave functions. 

Hiickel s application of this approach to the aromatic compounds gave new confidence to those physicists and chemists following up on the Hund-Mulliken analysis. It was regarded by many people as the simplest of the quantum mechanical valence-bond methods based on the Schrodinger equation. 66 Hiickel s was part of a series of applications of the method of linear combination of atom wave functions (atomic orbitals), a method that Felix Bloch had extended from H2+ to metals in 1928 and that Fowler s student, Lennard-Jones, had further developed for diatomic molecules in 1929. Now Hiickel extended the method to polyatomic molecules.67... 

The EPM required some measured data to determine the Fourier coefficients of the pseudopotential. However, the most modem approaches follow the Fermi [5] concept of developing a pseudopotential to yield a wave function without nodes that coincides with the all-electron atomic wave function outside the core and is still normalized. Several methods were developed [16-19] in the 1970s and 1980s, and new methods for constructing useful pseudopotentials continue to appear in the literature. The applications discussed here are mostly based on the pseudopotentials developed using the approach described in Ref. [19]. The important point to empha-... 

A quarter of a century ago the author stepped into Jens Oddershede s office and asked for support on a problem involving computation with atomic wave functions in connection with a new theoretical scheme to treat stopping of charged particles at intermediate speed. This visit resulted in two related publications, two joint papers and a number of follow-up studies by Jens and several others. In 1989 a Sanibel Symposium was devoted to aspects of the penetration of charged particles through matter, and since then, quite a few quantum chemists have joined the community of theoreticians dealing with particle penetration. 

The second and third terms involve excited states that produce electron correlation (particularly of the angular sort) in the closed 2s shells of the atoms. Therefore, the wave function for the asymptotic geometry is essentially the product of two atomic wave functions. 

Zener, C. (1930). Analytic atomic wave functions. Phys. Rev. 36, 51-56. 

---