Week 47: Coupled Cluster theory
Contents
Week 47, November 18-22, 2024
Introduction
The normal-ordered Hamiltonian
Hamiltonian
The reference state
Hartree-Fock basis
Short repetition: Practice in normal ordering
Answer
Hartree-Fock again
What does ``good'' mean?
Answer
To nuclear physics aficionadoss: How many nuclei are accessible with the coupled cluster method based on spherical mean fields?
Answer
The similarity transformed Hamiltonian
Clusters of excited states
Similarity transformation
What \( T \) leads to Hermitian \( \overline{H_N} \) ?
Answer
Understanding (non-unitary) similarity transformations
Answer
Non-hermitian operator
More formalism
Particle-hole excitations
How many unknowns?
Answer
Full clutser operator
Unknown aplitudes
Correlation energy
Good reference state
Exercise 1: Why is CCD not exact?
Computing the similarity-transformed Hamiltonian
The cluster operator
When does CCSD truncate?
Non-hermitian Hamiltonian
Similarity transformed Hamiltonian
Compute the matrix element \( \overline{H}_{ab}^{ij}\equiv \langle ij\vert \overline{H_N}\vert ab\rangle \)
Algebraic manipulations
Example: The contribution of \( [F, T_2] \) to \( \overline{H_N} \)
Antisymmetry
Final contribution
Assign the correct matrix element \( \langle pq\vert V\vert rs\rangle \) to each of the following diagrams of the interaction
CCD Approximation
Deriving the CCD equations
The algebraic expression
Computational cost of a CCD computation
Additional details
Factorizing the remaining diagrams of the CCD equation
Diagram 8
Diagram 9
Solving the CCD equations
More manipulations
CCD for the pairing Hamiltonian
Solving the CCD equations for the pairing problem
Python code
What \( T \) leads to Hermitian \( \overline{H_N} \) ?
What are the conditions on \( T \) such that \( \overline{H_N} \) is Hermitian?
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