Slater atomic wavefunctions

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. 

The understanding of electronic states in atoms is to a great extent based on Schrodinger s solution of the hydrogen-atom problem. These wavefunctions have the general form (Landau and Lifshitz, 1977)  

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  

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is 

At r—all the terms with r and r - are negligible. To take derivatives with respect to r, it is obvious that the derivative from the exponential factor of the Slater function, Eq. (6.2), is much larger than from the algebraic factor. Therefore, the Schrodinger equation implies 

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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... 

The normalisation factor is assumed. It is often convenient to indicate the spin of each electron in the determinant this is done by writing a bar when the spin part is P (spin down) a function without a bar indicates an a spin (spin up). Thus, the following are all commonly used ways to write the Slater determinantal wavefunction for the beryllium atom (which has the electronic configuration ls 2s ) ... 

McLean, A. D., and McLean, R. S. (1981). Roothaan-Hartree-Fock atomic wavefunctions. Slater basis set expansions for A=55-92. Atomic Data and Nuclear Data Tables 26, 197-401. 

For example, the ground state f 0 of the magnesium atom for which 12 electrons must be placed in the spin-orbitals lsOm% 2sOm% 2prn"s,3sOms is represented by the Slater determinantal wavefunction... 

The functions 4> are the spatial molecular (or atomic) orbitals or wavefunctions that (along with the spin functions) make up the overall or total molecular (or atomic) wavefunction ijj, which can be written as a Slater determinant (Eq. 5.12). Concerning the energies from the fact that... 

We have seen that the Slater determinant wavefunction for the ground state of the He atom may be written as a product of two parts, one factor containing only space functions,... 

The metals in Group 1 have one unpaired electron in the valence shell s orbital. This AO may be combined with a or p spin functions to form two spin orbitals. It is therefore possible to write down two Slater determinant wavefunctions for these atoms the ground states are doubly degenerate. Atoms that contain two or more unpaired electrons are more difficult to describe. For these species one is forced to form wavefunctions by linear combination of two or more Slater determinants. In the next section we shall deal with the simplest atom of this kind, viz. a helium atom excited to the 1x 25 electron configuration. 

One possibility as a basis set of functions to be used for/is the hydrogen-atom functions. As seen in the case of a helium atom, this is not a particularly good start. The hydrogen-atom wavefunctions do not account for shielding and other affects of the inter-electronic repulsion. A basis set of functions that take this into account is a much better starting point for the calculation. J. C. Slater created such a basis set of functions known as the Slater-type orbitals (STO). The functions have the following general form. 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

The ground state 9, (0 of the neutral target atom is represented by a Roothan-Hartree-Fock wavefunction that may be expanded in terms of Slater orbitals [45]. Denoting the triply differential cross section by... 

In analogy to using a linear combination of atomic orbitals to form MOs, a variational procedure is used to construct many-electron wavefunctions from a set of N Slater determinants y, i.e. one sets up a N x. N matrix of elements flij = (d>, H d>y) which, upon diagonalization, yields state energies and associated vectors of coefficients a used to define (fi as a linear combination of A,s ... 

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