_{1}

^{*}

The focus of our investigation is to evaluate one of the four contributing terms to the coulombic potential energy of an H
_{2} molecule. Specifically, we are interested in the term describing the electronic interaction of the charge distribution of one of the hydrogen atoms with the proton of the second atom. Quantum mechanics provides the charge distribution; hence, the evaluation of this term is a semi-classic quantum physics issue. For states other than the ground state the charge distributions are not spherically symmetric; they are functions of the radial and the angular coordinates. For the excited states we develop exact analytic expressions conducive to the potential energies. Because of the functional complexities of the wave functions, the evaluation of the core integrals is carried out utilizing symbolic capabilities of
Mathematica [1]. Plots of these energies vs. the distance between the two protons reveal global features.

Literature search reveals that quantum chemists customarily prefer adapting the shell method of charge distribution of the electron calculating the coulombic related potential energy issues of an H_{2} molecule. In this method one assumes that the charge distribution is composed of concentric spherical shells of finite thicknesses about the proton. Summing the potential contribution of the shells results in the total potential [_{2} molecule the author was unable to locate a reference addressing this approach. This approach is a one step- process and it is less cumbersome. As of our first objective in the analysis section we show the equivalency of these two methods. Our second objective is to adapt the latter method and craft an expression evaluating the electrostatic potentials of non-spherical charge distributions, i.e. distributions subject to states other than the ground state. The analysis section embodies one such expression enabling the evaluation of the potential of a desired excited state. In the course of derivation of this expression, because of the complexities of the wave functions we deploy the symbolic capabilities of Mathematica [

Following the objectives outlined in the previous section we consider the volume charge distribution method to evaluate the electrostatic potential of the charge distribution of a hydrogen atom. We consider two scenarios. First, we assume the distribution is confined within a sphere with a sharply defined radius R. For the case at hand the point of interest falls either outside or inside of the sphere; we will include more comments on this when we consider a diffused-edge sphere. We craft two distinct approaches evaluating their corresponding potentials. Second, because the charge distribution of the electron within the “sphere” according to the wave function description is diffused, we stretch the radius of the sphere harvesting the contribution of the entire charge distribution. We show for the diffused ground state the volume and the shell charge distribution methods are equivalent. Next, for the excited states following the same strategy we modify the calculation to include the features of the non-spherical distributions. Utilizing this formulation for selective cases explicitly we evaluate their corresponding potential energies as a function of the inner-atomic distance.

is the volume charge density, and

where, k is the electrostatic constant

where r_{<} (r_{>}) is the lesser (greater) of r and r', P_{ℓ}, are the Legendre polynomials [

er hand the spherical volume element is

For a special case of uniform charge density, (3) yields the classic expected potential

where q is the total charge within the sphere. That is the potential of a uniform charge distribution for exterior points, it is the same as if the entire charge within the sphere was concentrated at the center of the sphere.

Alternatively, one might omit the expansion given by (2) and directly integrate (1). To accomplish this, in (1) we substitute

ward integration yields (3). Although this method works for the exterior points, i.e. r > R, it fails for the interiors, r < R. For the latter we pursue the previous method; the details follow.

By splitting the integration in two radial zones (1) reads,

In Formulating (4) we apply the previous assumptions, that the distribution is spherically symmetric. Noting the orthogonality of the angular integration mentioned earlier, i.e. 2δ_{ℓ0}, (4) yields,

Assuming constant density, a straight forward integration of (5) yields,

Equation (6) for r = R is the expected potential; its value matches the potential of the exterior on the surface of the sphere. The potential is continuous across the boundary of the sphere.

The intent of developing and reviewing this formulation is to evaluate the potential of charge distribution of an atom. Noting the distribution has a diffused boundary and because the radius of the “sphere” for a diffuse surface theoretically is infinite, the potential at any point needs to be evaluated according to (5).

For instance the charge distribution of the ground state of the hydrogen atom is

where q_{e} is the electron charge, α = 2/a and a is the Bohr radius, respectively [

This is the same as [_{p}, yields the attractive coulomb potential energy of the electron-cloud of one of the atoms and the proton of the second atom in an H_{2} molecule.

For excited states of a hydrogen atom, density of the charge distribution is a function of {r,θ} and is independent of the azimuthal angle φ. It is,

here, R_{NL}(r) is the radial wave function and Y_{LM}(θ,φ) is the spherical harmonics [

In (9) although Y_{LM}’s are explicit functions of azimuthal angle φ’ their absolute squared are not; therefore, the integral over φ' gives a factor 2π. The “orbitals” are labeled by N = 1, 2, 3, and the L's are bound to_{ℓ}. _{ℓ}.

Inspecting

for ℓ; these are ℓ = 0 and ℓ = 0, 2, respectively.

cited state, only the even values of ℓ’s are conducive to non-zero

N | L | M | ||
---|---|---|---|---|

1 | 0 | 0 | 0 | |

2 | 0 | 0 | 0 | |

2 | 0 | 0 | 2 | 0 |

2 | 0 | 0 | 4 | 0 |

2 | 0 | −1 | 0 | |

2 | 0 | −1 | 2 | |

2 | 0 | −1 | 4 | 0 |

2 | 1 | 0 | 0 | |

2 | 1 | 0 | 2 | |

2 | 1 | 0 | 4 | 0 |

2 | 0 | 1 | 0 | |

2 | 0 | 1 | 2 | |

2 | 0 | 1 | 4 | 0 |

Applying (9) requires also knowing the explicit functional form of the radial wave functions. These functions are available in various references, e.g. [

Coulomb potential energies associated with various states of hydrogen atom are tabulated. The top left corner is the energy of the ground state. This is the same as discussed in [

N\L | 0 | 1 | 2 | 3 |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | ||||

4 |

V_{100 } | |
---|---|

V_{200 } | |

V_{210 } | |

V_{300 } | |

V_{310 } |

energies. The functional behavior of these plots is somewhat intuitive, meaning, the excited states vs. the ground state are more spacious; therefore, they have a longer impact range. These are vividly shown by the tails of their associated energy plots. A quick review of these graphs also reveals that the energies of the excited states associated with PE_{210}, PE_{310} and PE_{320} exhibit pronounced minima approximately at 1.2, 1 and 3.5 Å, respectively. Our ongoing related investigation focuses on the physics of these minima.

_{2} molecule depending to their excited state is about 1 Å to 2 Å, at short distances the coulomb energies are distinctly different. As shown on the right panel this distinction is more pronounced for L = 1 states.

Tabulated energy plots of

The left panel of

Motivation of proposing our investigation is to augment the current scope of energy issues of an H_{2} molecule. Charge distribution of hydrogen is a quantum physics concept; however, evaluation of the electrostatic potential

V_{211 } | |
---|---|

V_{311 } | |

V_{321 } | |

V_{322 } |

and potential energy associated with the charge distribution is a classic electrostatic problem. It is this semi- classic quantum physics nature of the issue that makes the problem appealing. Reviewing the sited references reveals that the majority of the investigations hover about the issues concerning the ground state charge distribution of hydrogen. It is natural to augment the investigation extending the analysis to issues beyond the ground state. Mathematical analysis of our present work differs from customary quantum chemists. However, we show for the ground state we are in agreement. Having established the equivalency of these two approaches we extend the formulation to consider states beyond the ground state. Because of the analytic complexities of the expressions we adapt the symbolic capabilities of Mathematica generating explicit analytic functions for the potential energies. Our work includes fresh graphic information not reported in various literatures; it puts the formulation to perspective. Currently we are extending the investigation to include electrostatic energies corresponding to the electron cloud-cloud interaction. Although this issue has been already investigated for the ground state configuration, there is no complete formulation to include the excited states. Pursuing our new initiative thus far, we have realized the complexities of the mathematical challenges. The author doubts that the electron exchange term of the coulomb energy of the excited states will ever be solved analytically!

A version of this investigation was presented at the ATINER International conference, Athens, Greece, July 2015.

Haiduke Sarafian, (2016) Coulomb Interaction in H_{2} Molecule for States beyond the Ground State. Journal of Modern Physics,07,528-535. doi: 10.4236/jmp.2016.76055