Differences

This shows you the differences between two versions of the page.

--- properties_and_expectation_values [2024/07/12 08:37] – external edit 127.0.0.1
+++ properties_and_expectation_values [2024/08/22 09:22] (current) – [Dipole fields (DIP)] peterk
@@ Line 291: / Line 291: @@
 ===== Finite field calculations =====
-Dipole moments, quadrupole moments etc. and the corresponding polarizabilities can be obtained as energy derivatives by the finite difference approximation. This is most easily done with the ''DIP'', ''QUAD'', or ''FIELD'' commands. An error will result if the added perturbation is not totally symmetric (symmetry 1). Note that the orbitals must be recomputed before performing a correlation calculation.
+Dipole moments, quadrupole moments etc. and the corresponding polarizabilities can be obtained as energy derivatives by the finite difference approximation. This is most easily done with the ''DIP'', ''QUAD'', or ''FIELD'' commands, which add a specified amount of the dipole, quadrupole, or any general, operators to the Hamiltonian. An error will result if the added perturbation is not totally symmetric (symmetry 1). Note that the orbitals must be recomputed before performing a correlation calculation.
 ==== Dipole fields (DIP) ====
-''DIP'',//xfield,yfield,zfield//;\\
+''DIP'',//dx,dy,dz//;\\
-''%%DIP+%%'',//xfield,yfield,zfield//;
+''%%DIP+%%'',//dx,dy,dz//;
+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.
-Add a finite dipole field to the one electron Hamiltonian and the core energy. The field strength is given by //xfield,yfield,zfield//. ''%%DIP+%%'' adds to any existing field, otherwise any previous field 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 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) ====
-''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) ====