Orbitals hydrogen-like

The wave mechanical treatment of the hydrogen atom does not provide more accurate values than the Bohr model did for the energy states of the hydrogen atom. It does, however, provide the basis for describing the probability of finding electrons in certain regions, which is more compatible with the Heisenberg uncertainty principle. Note that the solution of this three-dimensional wave equation resulted in the introduction of three quantum numbers (n, /, and mi). A principle of quantum mechanics predicts that there will be one quantum number for 

A little more difficulty is encountered with the conversion of the volume element dr = dx dy dz  

If the nucleus (which is placed in the center) has a positive charge of Ze, where e is the numerical value of the charge of the electron, we obtain the Hamiltonian operator for an electron in this central field, 

With V2 written out in polar coordinates, the Schrodinger equation for this system is 

As expected, the equation is a differential equation of second order in three variables. Let us try to guess a solution. 

This is substituted into the equation, and the r dependency is collected on the left side of the equation, the 6 and 0 dependency on the right  

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T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

Different Types of Complete Sets. Importance of the Continuum in Using Hydrogen-like Orbitals... 

The discrete hydrogen-like orbitals (nlm) are given by the formula ... 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

The radial wave functions used are thus the hydrogen-like 2p and 3d functions, J ai(r) and J 32-(r), for all orbitals of the L and M shells, respectively the symbols pts, and i 3j, piP, p3d represent these multiplied by the angular parts 1 (for s), /3 cos 8 (for p), and /5/4 (3 cos2 0-1) (for d), rather than the usual hydrogen-like orbitals. The 2-axis for each atom points along the internuclear axis toward the other atom. 

To evaluate the segmentary quantum numbers we observe from a comparison of equations (9) and (10) with the corresponding ones for a hydrogen-like orbit... 

The VB and the MO methods are rooted in very different philosophies of describing molecules. Although at the outset each method leads to different approximate wave functions, when successive improvements are made the two ultimately converge to the same wave function. In both the VB and MO methods, an approximate molecular wave function is obtained by combining appropriate hydrogen-like orbitals on each of the atoms in the molecule. This is called the linear combination of atomic orbitals (LCAO) approximation. 

In agreement with theoretical prediction, the experimental analysis shows the more positive atoms to be contracted. This is explained by the decrease in electron-electron repulsions, or, in a somewhat different language, the decreased screening of the nuclear attraction forces by a smaller number of electrons. This contraction is incorporated in Slater s rules for approximate, single exponential (and therefore nodeless), hydrogen-like orbital functions (Slater 1932). For a 2px orbital of a second-row atom, for example, the orbital function is given by... 

The term Q is difficult to evaluate quantitatively. It does not seem proper to use Slater nodeless orbitals since these are generally good only in the overlap region. If we use hydrogen-like orbitals for 2p and 2s, we obtain... 

In an atom with more than one electron, the energy of the 2p hydrogen-like orbitals is in fact a little higher than the energy of the 2s orbital, because Z in an atom that contains both 2s and 2p electrons is effectively larger for 2s than for 2p. This is partly due to the fact that the 2s electrons shield some of the positive charge of the nucleus from the 2p electrons. 

Because the Schrodinger equation cannot be solved exactly for polyelectron atoms, it has become the practice to approximate the electron configuration by assigning electrons to hydrogen-like orbitals. These orbitals are designated by the same labels as for hydrogen s orbitals and have the same spatial characteristics described in the previous section, Orbitals. ... 

Equation (2.1) cannot be solved exactly for a polyelectronic atom A because of complications resulting from interelectronic repulsions. We therefore use approximate solutions which are obtained by replacing A with a fictitious atom having the same nucleus but only one electron. For this reason, atomic orbitals are also called hydrogen-like orbitals and the orbital theory the monoelectronic approximation. 

In variance with the hydrogen-like and Slater functions the potential employed to formally construct the gaussian basis states has nothing to do with the real potential acting upon an electron in an atom. On the other hand the solutions of this (actually three-dimensional harmonic oscillator problem) form a complete discrete basis in the space of orbitals in contrast to the hydrogen-like orbitals. 

The problem now is to evaluate (6.27) for various choices of

ground state Is hydrogen-like orbitals which we calculated earlier in this chapter ... 

In the most commonly utilized approximation, the many-electron wave functions are written in terms of products of one-electron wave functions similar to the solutions obtained for the hydrogen atom. These one-electron functions used to construct the many-electron wave function are called atomic orbitals. They are also called hydrogen-like orbitals since they are one-electron orbitals and also because their shape is similar to that of the hydrogen atom orbitals. 

We will now examine the angular dependence of the hydrogen-like orbitals. In terms of spherical harmonics the solution to the Schrodinger equation may be written as... 

Here n denotes "effective quantum number", exponent 5C, is an arbitrary positive number, r, t, y) are polar coordinates for a point with respect to the origin A in which the function (2,3) is centered. Apart from the first two terms that represent a normalizing factor, the function (2,3) is closely related to hydrogen-like orbitals. For the hydrogen Is orbital the function I q q 0 identical with Q q, if we assume Z Z/n, However, it should be recalled that in contrast to hydrogen-like orbitals STO s are not mutually orthogonal. Another essential difference is in the number of nodes. Hydrogen functions have (n -i - 1) nodes, whereas STO s are nodeless in their radial part. Alternatively, the STO may be expressed by means of Cartesian coordinates as follows... 

GTO Gaussian-type orbital. Functions which differ from hydrogen-like orbitals in that the r dependence is exp(—ar ). 

STO Slater-Type Orbital. Functions which loosely resemble hydrogen-like orbitals, especially insofar as the dependence is exp(— r). 

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