Hydrogen atom orbital energy

For the hydrogen atom, orbital energy depends only on the value of n. For example, all four n = 2 orbitals have the same energy. All nine n = 3 orbitals have tbe same energy. What must the value of /be for each of these orbitals ... 

Within the hydrogen atom, the lower the value or n, the more stable will be the orbital. For the hydrogen atom, the energy depends only upon n for atoms with more than one electron the quantum number / is important as wel1. 

There are several discrete atomic orbitals available to the electron of a hydrogen atom. These orbitals differ in energy, size, and shape, and exact mathematical descriptions for each are possible. Following is a qualitative description of the nature of some of the hydrogen atomic orbitals. 

For a hydrogen atom the energy of the orbital is determined only by the principal quantum number, n, and n can take values 1,2, 3, and so on. This is the most fundamental division and is stated first in the description of an electron. The electron in a hydrogen atom is called Is1. The 1 gives the value of n the most important thing in the foremost place. The designation s refers to the value of . These two together, Is, define and name the orbital. The superscript 1 tells us that there is one electron in this orbital. 

In the hydrogen atom the energy is the same for the 2s and the three 2p states. In an atom with more electrons the energy is still the same for the three 2p states, but then the 2s state has a lower energy (penetrating orbits, p. 13). 

Fig. 1.1 Overlap of hydrogen atom orbitals to form (a) a bonding orbital, and (b) an antibonding orbital, with energies as shown in (c).

So far we have talked about the shapes of the hydrogen atomic orbitals but not about their energies. For the hydrogen atom the energy of a particular orbital is determined by its value of n. Thus all orbitals with the same value of n have the same energy—they are said to be degenerate. This feature... 

Identify the relationships among a hydrogen atom s energy levels, sublevels, and atomic orbitals. 

For the hydrogen atom, we can solve the Schrodinger equation exactly to obtain the allowed energy levels and the hydrogen atomic orbitals. The sizes and shapes of these orbitals tell us the probability distribution for the electron in each quantum state of the atom. We are led to picture this distribution as a smeared cloud of electron density (probability density) with a shape that is determined by the quantum state. 

For all other atoms, we have to generate approximations to solve the Schrodinger equation. The Hartree orbitals describe approximately the amplitude for each electron in the atom, moving under an effective force obtained by averaging over the interactions with all the other electrons. The Hartree orbitals have the same shapes as the hydrogen atomic orbitals—but very different sizes and energy values—and thus guide us to view the probability distribution for each electron as a smeared cloud of electron density. 

Atoms with many electrons are described by Hartree s SCF method, in which each electron is assumed to move under the influence of an effective field Veff (r) due to the average positions of all the other electrons. This method generates a set of one-electron wave functions called the Hartree orbitals ip (f) with energy values where a represents the proper set of quantum numbers. Hartree orbitals bear close relation to the hydrogen atomic orbitals but are not the same objects. 

Energies of the Hartree orbitals are different from those of the corresponding hydrogen atomic orbitals. For an atom with atomic number Z they can be estimated as -Zeff/n, where the effective nuclear charge experienced by each electron is determined by screening of that electron from the full nuclear charge by other electrons. 

An orbital is a volume of space about the nucleus where the probability of finding an electron is high. Unlike orbits that are easy to visualize, orbitals have shapes that do not resemble the circular paths of orbits. In the quantum mechanical model of the hydrogen atom, the energy of the electron is accurately known but its location about the nucleus is not known with certainty at any instant. The three-dimensional volumes that represent the orbitals indicate where an electron will likely be at any instant. This uncertainty in location is a necessity of physics. 

The principal quantum number, n, determines the energy of a hydrogenic atomic orbital through eqn 10.11. 

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