Electronic States and Potential Energy Functions Quiz

18 Questions

What role do the electronic energies, Ee(R), play in the nuclear motion?

Represent the potential energy functions experienced by the nuclei

What does the approximation for the nuclear problem involve?

Solving the Schrödinger equation for nuclear motion with electronic energies

What are En in the context of the electronic states?

Energy levels associated with nuclear vibrational and rotational motion

Why is the described scenario an approximation according to the text?

Neglecting electron-nuclei interactions

What does Term 1 in the full Schrödinger equation represent?

Potential energy due to electron-nuclei interactions

Which part of the equation is responsible for accounting for kinetic energy contributions?

Ĥnχn(R) = T̂n + Ee(R) χn(R)

What does the 1 ± SAB term in Ee± ensure in the context of bonding orbitals?

Antibonding orbitals are more antibonding than bonding orbitals are bonding.

How does increasing the number of basis functions in Ψe affect the LCAO approximation results?

It improves the accuracy towards the exact result.

What is a suggested way to improve basis functions in quantum mechanics?

Optimizing ξ to minimize the energy.

In the context of electronic states, what does the overlap density V(R) provide a qualitative guide to?

The nature of the binding and order of electronic states.

What is a significant advantage of using the variation principle in quantum mechanics?

Improving accuracy by optimizing coefficients to minimize energy.

How do atomic orbitals get labeled according to quantum numbers?

Using principal and orbital angular momentum quantum numbers.

What is the purpose of using the Born-Oppenheimer approximation in quantum mechanics?

To consider the motion of electrons and nuclei separately.

Which wavefunction describes the motion of the nuclei in the context of the given text?

χn(R)

In the context of quantum mechanics, what does the electronic Hamiltonian Ĥe represent?

A Hamiltonian for stationary nuclei.

What are adiabatic electronic states a result of in quantum mechanics?

Born-Oppenheimer approximation.

Which equation focuses on the electronic motion at fixed positions of the nuclei?

$Ĥe Ψe (r|R) = Ee (R)Ψe(r|R)$

In quantum mechanics, what factor allows for a separable product form of the total wavefunction?

Adiabatic separation

Study Notes

Schrödinger Equation for H2

The Schrödinger equation for H2 in the centre-of-mass (CM) frame is: Ĥ = T̂e + T̂n + V̂en + V̂nn + V̂ee
T̂e is the electronic kinetic energy, T̂n is the nuclear kinetic energy, V̂en is the electron-nucleus potential, V̂nn is the nucleus-nucleus potential, and V̂ee is the electron-electron potential

Born-Oppenheimer Approximation

The approximation assumes that electrons move much faster than nuclei and adapt quickly to changes in nuclear motion
The total wavefunction is written as a separable product: Ψ(r, R) = χn(R)Ψe(r|R)
χn(R) is the wavefunction describing the motion of the nuclei, and Ψe(r|R) is the electronic wavefunction (dependent on nuclear coordinates)

Electronic Hamiltonian and Adiabatic Separation

The electronic Hamiltonian is: Ĥe = Ĥ - T̂n
Solution of the simplified Schrödinger equation for electronic motion at fixed R yields a set of electronic states with energies Ee(R) that depend on nuclear positions
Ee(R) represents the potential energy functions (curves) experienced by the nuclei at a given separation, R

Nuclear Problem

For each electronic state, the Schrödinger equation for nuclear motion is: Ĥnχn(R) = T̂n + Ee(R) χn(R)
The electronic energies, Ee(R), play the role of the potential energy, V(R), experienced by the nuclei
Solution of the equation yields a set of energy levels, En, associated with vibrational and rotational motion of nuclei in a specific electronic state

Potential Energy Curves

Each electronic state is described by a different potential energy curve, with its own set of rotation-vibration states
The potential energy curves, V(R), include the Coulomb repulsion between the (fixed) nuclei

Limitations of the Born-Oppenheimer Approximation

The approximation is not exact, as the full Schrödinger equation is not separable
The approximation ignores the nuclear kinetic energy operator acting on the electronic wavefunction

Linear Combination of Atomic Orbitals (LCAO) Approximation

The LCAO approximation is used to simplify the electronic problem
The approximation has an error of ~8% in Re and ~46% in De for the ground electronic state of H2+
The accuracy can be improved by increasing the number of basis functions and optimizing the coefficients using the variation principle

Test your understanding of electronic states, potential energy functions, and the Schrödinger equation for nuclear motion. Explore how nuclei experience potential energy curves and the Coulomb repulsion between fixed nuclei.

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