Hydrogen-atom wave functions

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

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... 

In the general case valence-state functions rather than hydrogenic atomic wave functions feature in the bonding. It has been emphasized that bond formation consists of delocalization of an effective electron pair in the pres-... 

Hydrogen Atom Wave Functions Angular Functions Angular factors Real wave functions... 

Hydrogen Atom Wave Functions Radial Functions... 

If we identify with 2Zr/n at, where a0 = A2/4irVe2, then the functions Fy become identical with the hydrogen-atom wave functions for the value n = n of the principal quantum number n, but not for other values of n the functions F xM all contain the same exponential function of r, whereas the hydrogen-atom wave functions for different values of n contain different exponential functions of r. 

Let us suppose that for a given problem we have selected a certain set of wave functions which are known to represent the system in some extreme configuration. For the hydrogen molecule we might choose a set of hydrogen-atom wave functions which would represent... 

A list of the complete hydrogen atom wave functions is given in Table 22.2. 

CALCULATIONS WITH MODIFIED HYDROGEN ATOM WAVE FUNCTIONS... 

Source Hydrogen Atom Wave Functions Angular Functions, Physical Chemistry, 5th ed.,Gordon Barrow (c) 1988. McGraw-Hill Companies, Inc. 

Section 6.6 The Bound-State Hydrogen-Atom Wave Functions 143... 

FIGURE 6.10 Polar graphs of the 0 factors in the s and hydrogen-atom wave functions. 

Ket notation is conveniently used to specify eigenfunctions by listing their eigenvalues. For exan le, the hydrogen-atom wave function with quantum numbers n, I, m is denoted by = n/m). 

Consider what the H2 wave function would look like for large values of the inter-nuclear separation R. When the electron is neeu nucleus a, nucleus b is so far away that we essentially have a hydrogen atom with origin at a. Thus, when r is small, the ground-state electronic wave function should resemble the ground-state hydrogen-atom wave function of Eq. (6.104). We have Z = 1, and the Bohr radius Oq has the numerical value 1 in atomic units hence (6.104) becomes... 

The unperturbed system is two hydrogen atoms. We have two zeroth-order functions consisting of products of hydrogen-atom wave functions, and these belong to a degenerate level. The correct ground-state zeroth-order function is the linear combination (13.101). 

It is interesting to notice that when ) = 4, the number of linearly independent hyperspherical harmonics belonging to a given value of A is (A -t- 1), i.e., 1,4,9,16,.. and so on - exactly the same as the degeneracy of the solutions to the Schrodinger equation for a hydrogen atom. V. Fock was, in fact, able to show that the Fourier transforms of the hydrogen atom wave functions can be written in the form [27] ... 

Approximate wavefunctions for the bonding and antibonding states of can be obtained by combining hydrogen atomic wave-functions centred on the two nuclei. This is known as the linear combination of atomic orbitals (LCAO) method. 

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