# Molecular Hamiltonian - Ammonia

• I
• Konte
In summary, the conversation discusses the molecular Hamiltonian operator and its components, including kinetic and potential energy operators for electrons and nuclei. The topic of the "bi-stable" potential energy operator and its absence in the Hamiltonian for ammonia is brought up, with the explanation that it is a result of solving the electronic problem in the Born-Oppenheimer approximation. The "inversion coordinate" is also mentioned as a factor in the double-well potential of ammonia.
Konte
Hello everybody,

The general expression of molecular Hamiltonian operator for any molecule is:
$$\hat{H} = \hat{T}_n+\hat{T}_e+\hat{V}_{ee}+\hat{V}_{nn}+\hat{V}_{en}+\hat{f}_{spin-orbit}$$
where:
##\hat{T}## correspond to kinetic energy operator
##\hat{V}## correspond to potential energy operator
##e## and ##n## subscripts correspond to electrons and nucleus which compose the molecule.
My question is : when I try to apply this to the ammonia molecule case, it seems like incomplete for me because the famous "bi-stable" potential energy operator of ammonia is missing. How to introduce it?

Thank you everybody.

Konte.

Konte said:
My question is : when I try to apply this to the ammonia molecule case, it seems like incomplete for me because the famous "bi-stable" potential energy operator of ammonia is missing. How to introduce it?
What operator is that?

The double-well potential of ammonia comes about simply from solving the electronic problem with the Hamiltonian you gave, in the Born-Oppenheimer approximation. What you get then is a multidimensional surface, and there is a particular "cut" in that surface that corresponds to the "inversion coordinate," which is a double well.

Konte
DrClaude said:
What operator is that?
I mean to say : the double-well potential of ammonia.
You answered my question, I think.
Thanks.

Konte

## 1. What is a molecular Hamiltonian?

A molecular Hamiltonian is a mathematical representation of the total energy of a molecule, taking into account the positions and interactions of all its constituent atoms. It includes terms for kinetic energy, potential energy, and electron-nucleus interactions.

## 2. What is the significance of the ammonia molecule in molecular Hamiltonians?

Ammonia (NH3) is a commonly used molecule in quantum chemistry and molecular physics for studying the principles of molecular Hamiltonians. Its relatively simple structure allows for easier calculations and analysis compared to more complex molecules.

## 3. How is a molecular Hamiltonian for ammonia calculated?

The molecular Hamiltonian for ammonia is calculated using the Schrödinger equation, which describes the behavior of quantum particles. This equation is solved using a combination of experimental data and theoretical calculations to determine the energy levels and properties of the molecule.

## 4. What are the main components of a molecular Hamiltonian for ammonia?

A molecular Hamiltonian for ammonia includes terms for the kinetic energy of the atoms, potential energy from the interactions between the atoms and electrons, and electron-nucleus interactions. It also takes into account the vibrational and rotational motions of the molecule.

## 5. How is a molecular Hamiltonian for ammonia used in research and practical applications?

Molecular Hamiltonians for ammonia are used in a variety of research fields, such as quantum chemistry, molecular dynamics simulations, and spectroscopy. They are also used in practical applications, such as predicting the behavior and properties of ammonia in chemical reactions and industrial processes.

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