Fock operator for closed shell system

This leads to a doubling of the Coulomb operator, which is why it is multiplied by 2. The summation going up to N/2 takes into account the fact that there are only N/2 orbitals for N electrons, since each orbital can hold 2 electrons. In summary, the Fock operator for a closed system is given by f = H(core) + sum (2*f - k), where the summation goes up to N/2 and the Coulomb operator is multiplied by 2 due to the identical single-particle wavefunctions for two electrons in the same orbital.
  • #1
greisen
76
0
Hey,

The Fock operator for a closed system is given by

f = H(core) + sum (2*f - k)

where f is the Coulomb operator and k is the exchange operator. The summation goes from 1 to N/2. My questions is why is the summation to N/2 and why does one have to multiply the Coulomb operator by 2?

Thanks in advance

Best regards
 
Physics news on Phys.org
  • #2
N/2 is due to each orbital has to electrons hence 2*Coulombic.
 
  • #3
greisen said:
N/2 is due to each orbital has to electrons hence 2*Coulombic.
Correct. The spatial part of the single-particle wavefunctions for the two electrons in the same orbital are identical.
 

Related to Fock operator for closed shell system

What is the Fock operator for a closed shell system?

The Fock operator is a mathematical representation of the total energy of a closed shell system in quantum mechanics. It is used to describe the behavior of electrons within an atom or molecule, taking into account both the kinetic and potential energy of the electrons.

How is the Fock operator related to the Schrödinger equation?

The Fock operator is a key component of the Hartree-Fock method, which is a mathematical approach to solving the Schrödinger equation for multi-electron systems. It is used to construct the trial wavefunction, which is then refined through an iterative process until the lowest energy state is reached.

What is the significance of a closed shell system in quantum mechanics?

A closed shell system is one in which all electrons are paired and follow the Pauli exclusion principle, meaning they have opposite spin. This results in a more stable and symmetric electron configuration, making it easier to calculate the total energy of the system using the Fock operator.

How does the Fock operator account for electron-electron interactions?

The Fock operator includes terms that account for the repulsion between electrons, known as the Coulomb and exchange integrals. These integrals take into account the distance between electrons and their spin states, allowing for a more accurate calculation of the total energy of the system.

What are some limitations of the Fock operator for closed shell systems?

The Fock operator is based on the Hartree-Fock method, which makes several simplifying assumptions and approximations. This can lead to inaccuracies in the calculated energies, especially for systems with strong electron correlation. Additionally, the Fock operator does not account for relativistic effects, which may be important for heavy elements.

Similar threads

I In a closed loop system with a pump, how can we control the pressure?
    • Mechanics
Replies
26
Views
5K
B Solving for the Nth divergence in any coordinate system
    • Linear and Abstract Algebra
2
Replies
41
Views
3K
I Issues in understanding screening effects in the Jellium model
    • Atomic and Condensed Matter
Replies
0
Views
632
I Are Functions Really Equal? Investigating the Criteria for Function Equality
    • Calculus
Replies
7
Views
940
I What is the role of geometry and dimensional operations?
    • General Math
Replies
3
Views
447
I Very basic questions about operators in QM
    • Quantum Physics
Replies
31
Views
2K
Bounded operators on Hilbert spaces
    • Calculus and Beyond Homework Help
2
Replies
43
Views
3K
Work done by Octane Combustion Engine
    • Biology and Chemistry Homework Help
Replies
2
Views
1K
A Numerical Hartree Fock with Finite Difference Matrices for Helium
    • Atomic and Condensed Matter
Replies
4
Views
2K
B Looking for context for a textbook quote
    • General Math
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top