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properties_and_expectation_values [2024/08/21 16:06] – [Dipole fields (DIP)] peterkproperties_and_expectation_values [2024/08/22 09:22] (current) – [Dipole fields (DIP)] peterk
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 ==== Dipole fields (DIP) ==== ==== Dipole fields (DIP) ====
  
-''DIP'',//xfield,yfield,zfield//;\\ +''DIP'',//dx,dy,dz//;\\ 
-''%%DIP+%%'',//xfield,yfield,zfield//;+''%%DIP+%%'',//dx,dy,dz//;
  
-Add a finite combination of dipole operators to the Hamiltonian (both the 1-electron operator and the core energy). The combination is given by //xfield,yfield,zfield//. ''%%DIP+%%'' adds to any existing field, otherwise any previous field is removed.+Add a finite combination of the dipole operators $\vec\mu=(\mu_x, \mu_y, \mu_z)$, $H_1=\vec d \cdot \vec \mu$ ( $\vec d=(\textit{dx},\textit{dy},\textit{dz})$)  to the Hamiltonian (both the 1-electron operator and the core energy). ''%%DIP+%%'' adds to any existing field, otherwise any previous perturbation is removed.
  
-The perturbed hamiltonian represents a physical system in a uniform electric field with electric field strength $\vec F= -(xfield,yfield,zfield)$.+The perturbed hamiltonian represents a physical system in a uniform electric field with electric field strength $\vec F= -\vec d$. Therefore the corresponding energy-derivative form of the dipole moment projection in this direction can be obtained as $$|\vec F|^{-1}\vec F \cdot \vec \mu  = |2\vec d|^{-1}(E(\vec d)-E(-\vec d)) + O(|\vec d|^2)= |\vec d|^{-1}(E(\vec d)-E(\vec 0)) + O(|\vec d|).$$ 
 +The diagonal polarisability in this direction can similarly be calculated via 
 +$$\alpha_{\vec d, \vec d}  = |\vec d|^{-2}(E(\vec d)+E(-\vec d)-2E(\vec 0)) + O(|\vec d|^2).$$
  
 ==== Quadrupole fields (QUAD) ==== ==== Quadrupole fields (QUAD) ====
  
-''QUAD'',//xxfield,yyfield,zzfield,xyfield,xzfield,yzfield//;\\ +''QUAD'',//qxx,qyy,qzz,qxy,qxz,qyz//;\\ 
-''%%QUAD+%%'',//xxfield,yyfield,zzfield,xyfield,xzfield,yzfield//;+''%%QUAD+%%'',//qxx,qyy,qzz,qxy,qxz,qyz//;
  
-Exactly as the ''DIP'' command, but adds a quadrupole field.+Exactly as the ''DIP'' command, but adds a combination of quadrupole operators to the Hamiltonian.
  
 ==== General fields (FIELD) ==== ==== General fields (FIELD) ====