Kimia Kuantum - 08B Atom Berelektron Banyak :: Helium (Bag. 1)

Summary

This detailed script explores the quantum mechanics of the helium atom, focusing on the Schrödinger equation and the challenges posed by electron-electron interactions. It explains the approximation of the helium atom's wave function, where simplifications are made by neglecting the electron-electron repulsion. The variational method is introduced to improve the accuracy of energy calculations by adjusting the effective nuclear charge (Z) to account for shielding effects. The resulting energy closely matches experimental data, significantly reducing errors and demonstrating the importance of sophisticated methods in quantum systems with multiple electrons.

Takeaways

• 😀 The helium atom consists of two electrons and has a nuclear charge of +2.
• 😀 The Schrödinger equation is used to describe the behavior of electrons, but electron-electron interaction makes exact solutions difficult.
• 😀 The Hamiltonian operator for helium includes the kinetic and potential energies of both electrons, as well as the interaction between them.
• 😀 A common approximation is to neglect electron-electron interactions and treat the electrons independently, similar to the hydrogen atom model.
• 😀 When electron-electron interactions are ignored, the energy of the helium atom can be approximated by summing the individual electron energies.
• 😀 The Schrödinger equation for the helium atom can be simplified into two separate terms for each electron’s energy if electron-electron interaction is neglected.
• 😀 Using a variational method with an adjustable parameter (Z or beta), more accurate results for the helium atom’s energy can be obtained.
• 😀 The variational method reduces errors in energy calculation, with the best parameter providing results close to experimental data.
• 😀 Initial calculations without considering electron-electron repulsion resulted in a significant error of around 30%.
• 😀 The effect of shielding occurs when one electron reduces the effective nuclear charge seen by the other, leading to an effective charge of 1.68 instead of 2 for each electron in helium.

Q & A

What is the atomic structure of helium as discussed in the transcript?

Helium consists of two protons, two neutrons, and two electrons. The electrons are in the 1s orbital, and the atomic number (Z) is 2, meaning it has a +2 positive charge at its core.

Why can the Schrödinger equation for the helium atom not be solved exactly using the separation of variables approach?

The Schrödinger equation for the helium atom cannot be solved exactly due to the presence of electron-electron interaction, which depends on the positions of both electrons. This interaction complicates the separation of variables approach, making it impossible to split the equation into simpler parts.

What happens if the electron-electron interaction term is ignored in the helium atom's Hamiltonian?

If the electron-electron interaction term is ignored, the Hamiltonian can be approximated as the sum of two independent single-electron Hamiltonians. This leads to a simpler solution for the wave function, but it significantly overestimates the atom's energy.

How is the helium atom's wave function approximated in this method?

The wave function of the helium atom is approximated by the product of two hydrogen-like wave functions, one for each electron, each in the 1s orbital. This approximation simplifies the calculation but does not perfectly represent the true wave function.

What is the calculated energy of the helium atom in the ground state using the simplified approach, and how does it compare to experimental values?

Using the simplified approach, the ground state energy of the helium atom is calculated to be -18.8 eV. This is significantly higher than the experimentally determined value of approximately -79.0 eV, leading to an error of around 30%.

What role does the variational method play in improving the energy approximation for the helium atom?

The variational method improves the energy approximation by introducing a parameter (Zeta) in the wave function to better match the actual behavior of the atom. This method helps minimize the energy difference between the approximated and actual values by adjusting the parameter to optimize the energy calculation.

What was the value of Zeta that minimized the energy in the variational method, and how did it affect the results?

The value of Zeta that minimized the energy in the variational method was 1.6875. This adjustment significantly improved the energy calculation, reducing the error from 30% to just about 1.5%, bringing the result much closer to the experimental value.

How does the Zeta parameter affect the perceived charge of the nucleus as seen by the electrons?

The Zeta parameter reduces the effective nuclear charge that each electron experiences. In the case of helium, the electrons perceive a charge of about +1.68 instead of the full +2, due to the shielding effect caused by the presence of both electrons in the same orbital.

What is the shielding effect mentioned in the transcript, and how does it impact the helium atom?

The shielding effect occurs because the electrons in the inner orbital reduce the effective charge experienced by the outer electron. In helium, the two electrons in the 1s orbital shield each other from the full nuclear charge, leading to a perceived charge of 1.68 rather than the expected 2.

How does the variational method compare to the original approximation in terms of accuracy for the helium atom's energy?

The variational method significantly improves the accuracy of the energy calculation for the helium atom. While the original approximation, which ignored the electron-electron interaction, gave an error of around 30%, the variational method reduced this error to only about 1.5%, providing a much more accurate result.