Orbitals Laguerre

These functions are universally known as Slater type orbitals (STOs) and are just the leading term in the appropriate Laguerre polynomials. The first three Slater functions are as follows ... 

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

Atomic Size The associated Laguerre polynomial (x) is a polynomial of degree nr = n — l — 1, which has nr radial nodes (zeros). The radial distribution function therefore exhibits n — l maxima. Whenever n = l + 1 and the orbital quantum number, l has its largest value, there is only one maximum. In this case nT = 0 and from (14) follows... 

The natures of the Lanczos functions and various properties of Stleltjes orbitals are best demonstrated by specific example. In Figure 3 are shown the first ten radial Lanczos functions for the ls->kp spectrum of a hydrogen atom (18). These are obtained from solution of Equations (4) for j=l to 10 employing the appropriate hydrogenic Hamiltonian in the operator A(H) and the Is ground state orbital in the test function. In this case. Equation (4b) can be solved for the v. in terms of Laguerre functions with constant expo-... 

The exact solutions to the separate equations, which result from this coordinate transformation of the Schrddinger equation for the hydrogen atom, are the sets of functions known as the associated Laguerre polynomials, for the radial equation, and the spherical harmonics, for the angular equation. The quantum numbers, n,l and m arise naturally in the solution of Schrddinger s equation, and so the symbolic form, for the eigenfunction solutions to the H-atom problem, known as atomic orbitals, is... 

The Is atomic orbital for the hydrogen atom results as an exact solution, for the choice of the first Laguerre polynomial (n = 1) for the radial wave function and the lowest spherical harmonic (/ = 0) Yqo, for the angular wave function. Thus, from Table 1.1, the normalized Is atomic orbital for the hydrogen atom is. 

FIGURE 3.7 The representations of the electronic probability density of existence (wave-functions) for Hydrogenic atoms, for the superior (excited) levels (or shells, quantified by the number n) with the respective sub-sheUs (or orbitals, quantified by the mixed numbers nl), employing the derived radial wave functions in terms of respective Laguerre polynomials of Table 3.1. 

We put r, where A is a natural number or zero, instead of Laguerre polynomials given on p. 179. The second difference is in the orbital exponent, which has no constraint except that it has to be positive. ... 

This equation does not depend on the m quantum number since the single electron is assumed to move in a central field. The radial functions are thus the same for orbitals, which are different only in m. A number of known sets of functions satisfy the mentioned boundary conditions. The Laguerre functions are known to be solutions of Equation 2.13. R(0) has to be finite at the nucleus and R(r) 0 exponentially when r oo, and this is, in fact, satisfied by Laguerre functions that are of the following type ... 

The lowest power in a Laguerre polynomial is n - (number of terms) = if. Thus, all s orbitals have a constant term in the polynomial, that is R(0) 0 all p orbitals are linear in r for small r all d orbitals are quadratic in r for small r, etc. AO with if = 1 cannot have a constant term in the polynomial in Equation 2.14, since Equation 2.13 is then unsatisfied. AO with if = 2 also cannot have a linear term. The radial functions are thus summarized in Table 2.1. 

Slater derived the screening rules from empirical data and these rules work well, down to the transition metal series. The hydrogen-like 3d and 4f orbitals are described by a single term in the Laguerre polynomial, but this simple description is not favorable when penetration and screening are included. The 3d and 4f orbitals have an inner part where screening is less important than in the outer parts. At least two exponential functions are needed for a relevant description. 

L total electronic orbital angular momentum, associated Laguerre polynomial (C )... 

Thus we have demonstrated how the L j /(p) polynomials can be generated and that they do satisfy the general associated Laguerre polynomial equation. Schrodinger worked out the Hydrogen orbitals from these functions in his third revolutionary paper [7] and perhaps we can appreciate the patience required to carry the derivation to useful results ... 

There is a polynomial solution for the radial part of the H atom solution of the Schrodinger equation. The functions are related to previously studied Laguerre polynomials. The total solution for the H atom is the product of the rigid rotor (0, < ) wave functions with the Laguerre radial functions. The eigenvalues for the orbitals are exactly the same as for the... 

The Laguerre radial solutions form an orthonormal set of functions and the orthogonality of the various members of the set is accomplished using a system of (n i l) radial nodes (zero crossings in the r-coordinate). The 2s orbital has a small part like the Is in it... 

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