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--- properties_and_expectation_values [2024/08/21 16:28] – [Dipole fields (DIP)] peterk
+++ properties_and_expectation_values [2024/08/22 09:22] (current) – [Dipole fields (DIP)] peterk
@@ Line 300: / Line 300: @@
 ''%%DIP+%%'',//dx,dy,dz//;
-Add a finite combination $H_1=\vec d \cdot \vec \mu$, $\vec d=(\textit{dx},\textit{dy},\textit{dz})$ of dipole operators to the Hamiltonian (both the 1-electron operator and the core energy). ''%%DIP+%%'' adds to any existing field, otherwise any previous perturbation 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 $\vec F= -\vec d$. 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 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) ====