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documentation:language_reference:functions:potentialexpandedonclm [2018/01/10 23:20] – Maurits W. Haverkortdocumentation:language_reference:functions:potentialexpandedonclm [2025/11/20 03:29] (current) – external edit 127.0.0.1
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 Given the onsite energies of the orbitals of all possible irreducible representations a potential expanded on renormalised Spherical Harmonics is created.  Given the onsite energies of the orbitals of all possible irreducible representations a potential expanded on renormalised Spherical Harmonics is created. 
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 +Within crystal field or ligand field theory it is common practice to expand a potential on Spherical Harmonics. A potential expanded as such can be used to create a crystal field operator with the function NewOperator("C", ...). For more information see the documentation of the crystal field [[documentation:standard_operators:crystal_field|operator]]. 
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 ===== Output ===== ===== Output =====
  
-  * Akm : Table containing {{k, m, Coefficient}, {k, m, Coefficient}, ...}+  * Akm : Table containing "$\{\{k_1,m_1,A_{k_1,m_1}\},\{k_2,m_2,A_{k_2,m_2}\},...\}$" defining the angular part of the potential such that $\big\langle \varphi_{\tau_1}(\vec{r}) \big| V(\vec{r}) \big| \varphi_{\tau_2}(\vec{r}) \big\rangle = \sum_{k=0}^{\infty}\sum_{m=-k}^{k}  A_{k,m} \big\langle Y_{l_1,m_1} \big| C_{k,m} \big| Y_{l_2,m_2} \big\rangle$, with $\varphi_{\tau_1}(\vec{r})=R_{n_1,l_1}(r) Y_{l_1,m_1}(\theta\phi)$
  
 ===== Example ===== ===== Example =====
  
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-Within crystal field or ligand field theory it is common practice to expand a potential on Spherical Harmonics. A potential expanded as such can be used to create a crystal field operator with the function NewOperator("C", ...). For more information see the documentation of the crystal field [[documentation:standard_operators:crystal_field|operator]]. +The following example creates a potential expanded on renormalised spherical Harmonics for different point-groups and prints the expansion as well as the Hamiltonian for an p, d and f shell that arises from this potential. The energies of the orbitals of a given irreducible representation are set to arbitrary values. 
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 +The example firstly defines three functions (for the p, d and f shell), that given a potential expanded on spherical harmonics print this potential, create an Hamiltonian on the basis of Spherical Harmonics and creates the Hamiltonian on a basis of Kubic Harmonics. The crystal field Hamiltonians are printed as a matrix, the basis functions used for this matrix are printed above the matrix. 
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