Write kinetic energy part of the Hamiltonian operator for H2 molecule.
Understand the Problem
The question is asking about the kinetic energy component of the Hamiltonian operator for a hydrogen molecule (H2). This involves understanding quantum mechanics and specifically how the Hamiltonian operator is constructed in this context.
Answer
- (ħ² / 2mₑ) ∇₁² - (ħ² / 2mₑ) ∇₂² - (ħ² / 2M₁) ∇₁ₙ² - (ħ² / 2M₂) ∇₂ₙ²
The kinetic energy part of the Hamiltonian operator for the H2 molecule involves the kinetic energy of the nuclei and the electrons, represented as:
T = - (ħ² / 2mₑ) ∇₁² - (ħ² / 2mₑ) ∇₂² - (ħ² / 2M₁) ∇₁ₙ² - (ħ² / 2M₂) ∇₂ₙ².
Here, ħ is the reduced Planck's constant, mₑ is the mass of the electron, M₁ and M₂ are the masses of the two nuclei, and ∇ represents the Laplacian operator.
Answer for screen readers
The kinetic energy part of the Hamiltonian operator for the H2 molecule involves the kinetic energy of the nuclei and the electrons, represented as:
T = - (ħ² / 2mₑ) ∇₁² - (ħ² / 2mₑ) ∇₂² - (ħ² / 2M₁) ∇₁ₙ² - (ħ² / 2M₂) ∇₂ₙ².
Here, ħ is the reduced Planck's constant, mₑ is the mass of the electron, M₁ and M₂ are the masses of the two nuclei, and ∇ represents the Laplacian operator.
More Information
This expression includes contributions from both electronic and nuclear kinetic energies, fundamental to understanding the total energy in quantum molecular systems.
Tips
Be careful to distinguish between the Laplacians for electrons and nuclei. Also, correctly apply atomic units if specified.
Sources
- Lecture 2 Hamiltonian operators for molecules CHEM6085 - southampton.ac.uk
- Quantum Chemistry 10.1 - Hydrogen Molecule Hamiltonian - YouTube - youtube.com
- 5.2 The Hydrogen Molecule - Florida State University - web1.eng.famu.fsu.edu
