Atom orbital calculations

🧪 Schrödinger Equation and the 2p Orbital

The 2p orbital is one of the solutions to the Time-Independent Schrödinger Equation (TISE) for a central Coulomb potential. Even though the full solution is exact only for hydrogenic atoms (1 electron), it provides the conceptual and mathematical basis for understanding multi-electron atoms like oxygen via approximations.

The axially symmetric dumbbell solution for the 2p orbitals are the simplest form for a self-sustained resonance.

🔬 Time-Independent Schrödinger Equation (Hydrogen-like Atom)

$$ -\frac{\hbar^2}{2\mu} \nabla^2 \psi(r, \theta, \phi) - \frac{Ze^2}{4\pi\varepsilon_0 r} \psi(r, \theta, \phi) = E \psi(r, \theta, \phi) $$

• $ \psi $: wavefunction
• $ \mu $: reduced mass
• $ Z $: atomic number (Z=8 for O, Z=1 for H-like model)
• $ \nabla^2 $: Laplacian in spherical coordinates

Separation of Variables

$$ \psi(r, \theta, \phi) = R_{n\ell}(r) \cdot Y_\ell^m(\theta, \phi) $$

• $ R_{n\ell}(r) $: radial function
• $ Y_\ell^m $: spherical harmonic (angular function)
• $ n = 2, \ell = 1, m = -1, 0, +1 $ for 2p orbital

Radial Part (Hydrogenic 2p Orbital)

$$ R_{21}(r) = \frac{1}{\sqrt{3}} \left( \frac{Z}{a_0} \right)^{3/2} \cdot \left( \frac{r}{a_0} \right) e^{-Zr/2a_0} $$

• $ a_0 $: Bohr radius
• Exponential decay with radial node at $ r = 0 $

Angular Part (p-Orbitals)

• $ Y_1^0(\theta, \phi) \propto \cos\theta $ → 2p$_z$
• Real combinations of $ Y_1^{\pm1} $ yield 2p$_x$, 2p$_y$
• Shape: Dumbbell, directionally oriented

🧾 In Multi-electron Atoms (e.g. Oxygen)

• Use effective nuclear charge $ Z_{\text{eff}} < Z $ to account for shielding
• 2p orbital shape retained, but energy and size shift

🔬 Solution of the Equation for the Olavian p-orbital

We aim at a solution for a custom Orbital with Circular XY Projection & Dumbbell XZ/YZ Projections

🧠 Goal

Construct a solution to the time-independent Schrödinger equation that:

• Appears dumbbell-like in the xz and yz planes (reflecting 1s orbital radius)
• Appears circularly symmetric in the xy plane (reflecting 2s orbital radius)

Base Hydrogenic 2p Orbitals

2p orbitals in Cartesian form (from spherical harmonics and radial functions):

• $ \psi_{2p_x} \propto x \cdot R(r) $
• $ \psi_{2p_y} \propto y \cdot R(r) $
• $ \psi_{2p_z} \propto z \cdot R(r) $

Where:

• $ R(r) \propto r \cdot e^{-Zr / 2a_0} $ is the radial part
• $ r = \sqrt{x^2 + y^2 + z^2} $

Composite Orbital Construction

Use the complex superposition:

$$ \psi(\vec{r}) = \frac{1}{\sqrt{2}} \left( \psi_{2p_x} + i \psi_{2p_y} \right) = \frac{1}{\sqrt{2}} (x + i y) \cdot R(r) $$

• Equivalent to spherical harmonic $Y_1^{+1}$
• Probability density: $|\psi|^2 = \frac{1}{2}(x^2 + y^2) \cdot R(r)^2$

🔍 Projection Properties

✅ Conclusion

The orbital: $$ \psi = \frac{1}{\sqrt{2}}(x + i y) \cdot r \cdot e^{-r / 2} $$ yields dumbbell-shaped projections in xz and yz planes, and a circular projection in the xy plane, fulfilling the desired spatial symmetry via constructive superposition of p orbitals.

🧪 Standard sp² Hybrid Orbital — Quick Summary

Why was this orbital introduced in standard physics? This soluation also provides a more or less planar symmetric version needed to explain the 120° bonds of atoms e.g. carbon.

A hybridiced $sp^2$ shell seems no to be the solution we need for a rotonal Olavian orbital. But as you have seen in the Olavian Atom model, we will come up with a two atom rotonal resonance solution for the Schrödinger equations accounting for the two nucleus situation. It will show a kind of 120°-ish shape too. At least if projected onto a plane.

✅ Why It’s Needed

The sp² orbital = 1 s orbital + 2 p orbitals (typically $p_x$ and $p_y$) was introduced to explain:

• Trigonal planar geometries with three equivalent orbitals arranged 120° apart in a plane
• Equal bond energies in molecules like ethene (C₂H₄), BF₃
• Bonding in atoms with three electron domains

Classical s and p orbitals cannot account for these bonding patterns — hybridization solves this.

When It’s Used

• Atoms forming double bonds → σ bond from sp², π bond from p_z
• Trigonal planar molecules (e.g. Ethene (C₂H₄), Boron trifluoride (BF₃), Formaldehyde (H₂CO)
• Lone pairs can also occupy sp² orbitals

🎨 What It Looks Like

• Asymmetric lobes: one large lobe (bonding), one small lobe (antibonding)
• All lie in the same plane, spaced at 120°
• Capable of σ bonding through end-to-end orbital overlap
• Remaining p_z orbital lies perpendicular to the plane (used in π bonding)

👉 ChemLibreTexts – sp² Hybrid Orbital Visualization

🧾 Conclusion

The sp² hybrid orbital enables planar, directional σ bonding with 120° angles — essential for understanding molecules with double bonds or trigonal planar geometry.

🔬 Superposition of Two Orbitals in Schrödinger Theory

Looking at atoms with multiple electrons we smoothly leave the field of single excited states/orbitals of a “simple” hydrogen atom. We enter a world, where stable orbitals might arise not as an “eigenstate” of a single wave-function (self-sustained) but as a well-harmonic compound (superposition) of two (or more) electron trajectories.

🧠 Core Question

Can two separated, oscillating orbital components form a resonant, stable solution to the Schrödinger equation — resembling a wobbling ring or multi-lobed orbital?

✅ Schrödinger Equation Basics

The time-independent Schrödinger equation for a single particle:

$$ \hat{H} \psi(\vec{r}) = E \psi(\vec{r}) $$

• Solutions (orbitals) must be eigenfunctions of the Hamiltonian.
• Superpositions of eigenfunctions are allowed if they share the same energy or lie in a degenerate space.

Superposition in Practice

You can write a superposed orbital as:

$$ \psi(\vec{r}) = c_1 \psi_A(\vec{r}) + c_2 \psi_B(\vec{r}) $$

• ( \psi_A ), ( \psi_B ): localized in different spatial regions.
• Valid only if ( \psi ) satisfies ( \hat{H} \psi = E \psi ).
• Otherwise, ( \psi ) is not an eigenfunction, but may still represent a time-dependent or approximate state.

Going Beyond: Resonant Lobes & Coupled Trajectories

To model mutually stabilizing lobes or spatial oscillation asymmetries, we need:

Option 1: Superposition of atomic orbitals

• E.g., combine ( 2s + 2p ) or ( 1s + 3d )
• Requires careful phase and energy matching

Option 2: Multi-center solutions

• E.g., molecular orbitals in H₂⁺ or N₂
• Constructed using LCAO (Linear Combination of Atomic Orbitals)
• Electron delocalizes between centers in resonance-stabilized states

Option 3: Multi-electron systems

• Use the many-body Schrödinger equation:

$$ \hat{H} \Psi(\vec{r}_1, \vec{r}_2) = E \Psi(\vec{r}_1, \vec{r}_2) $$

• Includes electron-electron interaction
• Solved using Hartree-Fock, DFT, or CI methods

🧾 Conclusion

✅ Schrödinger’s equation can support spatially structured superpositions, but:

• True resonant coupling and stabilization between lobes require: Degenerate orbitals, or Multi-center or multi-electron formulations

Such complex shapes often emerge in molecular orbitals, delocalized systems — not in isolated hydrogenic atoms.

Authors comment: did you notice the standard atom models base on the excited eigenstates of a simple hydrogen atom? All atoms are mathematically built up as an additive superposition of independent H-Atoms and not as a compound harmonic of multiple electrons. This avoids the need to model inter-electron resonances, as long as this matches to the experimental results good enough.