Recent changes to Demos.ConicalInter.Main

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 31 May 2018 14:14:18 -0000

--- v28
+++ v29
@@ -27,6 +27,7 @@
 [[img src=Saddles.gif alt="\rho_S=\frac{\kappa}{\omega+g},\,\phi_S=\pi/3,\pi,5\pi/3,\,E_S=-\frac{\kappa^2}{2(\omega+g)}"]]

 * see e.g. [work by Zwanziger and Grant](http://dx.doi.org/10.1063/1.453083)
+* see e.g. [work by Viel and Eisfeld](http://dx.doi.org/10.1063/1.1646371)
 * see e.g. [work by Eisfeld and Viel](http://dx.doi.org/10.1063/1.1904594)

 Linear E x e Jahn-Teller system: Double cone potentials

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Mon, 07 May 2018 12:47:40 -0000

--- v27
+++ v28
@@ -1,12 +1,12 @@
 WavePacket demo example: Generic conical intersections
 ======================================================

-Once considered to be of merely academic interest, by now conical intersections of potential energy surfaces are known to be of paramount importance for many processes in photophysics, photochemistry, and photobiology. The most elementary, generic case of a conical intersection arises in molecular physics is the socalled case of E x e Jahn Teller coupling. It arises when a molecule with (at least!) threefold rotational symmetry is in a degenerate electronic state (E irrep), coupled to a pair of degenerate nuclear distortions of the same symmetry (e irrep). The result of the Jahn Teller theorem stating the incompatibility of degenerate (electronic) states and symmetric (nuclear) configurations predicts spontaneous symmetry breaking for all non-linear molecules.
+Once considered to be of merely academic interest, by now conical intersections of potential energy surfaces are known to be of paramount importance for many processes in photophysics, photochemistry, and photobiology. The most elementary, generic case of a conical intersection arises in molecular physics is the socalled case of E x e Jahn Teller coupling. It arises when a molecule with (at least) threefold rotational symmetry is in a degenerate electronic state (E irrep), coupled to a pair of degenerate nuclear distortions of the same symmetry (e irrep). The result of the Jahn Teller theorem stating the incompatibility of degenerate (electronic) states and symmetric (nuclear) configurations predicts spontaneous symmetry breaking for all non-linear molecules.

 Formalism
 ---------

-Diabatic potential energy matrix (Cartesian coordinates)
+Diabatic potential energy matrix (normal coordinates)

 [[img src=Diabatic.gif alt="V(x,y)=\frac{\omega}{2}\begin{pmatrix}x^2+y^2&0\\0&x^2+y^2\end{pmatrix}+\kappa\begin{pmatrix}x&y\\y&-x\end{pmatrix}+\frac{g}{2}\begin{pmatrix}x^2-y^2&-2xy\\-2xy&y^2-x^2\end{pmatrix}"]]

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 23 Feb 2017 12:38:49 -0000

--- v26
+++ v27
@@ -42,7 +42,7 @@
 Spin-Boson system in 1D
 ---------------------

-In a diabatic representation, a one-dimensional spin-boson Hamiltonian can be represented by two harmonic potentials displaced in position and energy and a constant coupling between the two states. In the adiabatic representation, the two potential energy curves exhibit an avoided crossing. The corresponding quantum dynamics becomes incresingly complicated as time goes by: Upon each passage of the avoided crossing, each wavepacket spwns another one, eventually leading to rather complicated interference patterns.  [Learn more ...](Demos.ConicalInter.OneDim)
+In a diabatic representation, a one-dimensional spin-boson Hamiltonian can be represented by two harmonic potentials displaced in position and energy and a constant coupling between the two states. In the adiabatic representation, the two potential energy curves exhibit an avoided crossing. The corresponding quantum dynamics becomes incresingly complicated as time goes by: Upon each passage of the avoided crossing, each wavepacket spawns another one, eventually leading to rather complicated interference patterns.  [Learn more ...](Demos.ConicalInter.OneDim)

 Geometric phase effects
 -----------------------

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 23 Feb 2017 12:33:56 -0000

--- v25
+++ v26
@@ -34,7 +34,7 @@

 In linear approximation, i.e., for vanishing quadratic (and higher) terms (κ ≠ 0, ω=g=0), the two adiabatic potential energy surfaces of the corresponding Jahn-Teller system can be represented by a pair of funnels forming a linear conical intersection. A wavepacket touching the catchment area of the intersection is subject to strong nonadiabatic effects. [Learn more ...](Demos.ConicalInter.Linear)

-Spin-Boson system
+Spin-Boson system in 2D
 -----------------

 If harmonic potentials are added to the linear E x e Jahn-Teller system, the prototypical spin-boson Hamiltonian is obtained with quadratic potentials on the diagonals and linear terms on the off-diagonal entries of the (diabatic!) potential matrix (κ ≠ 0, ω ≠ 0, g=0). Due to the multiple passing of high energy wavepackets of the conical intersection, the quantum dynamics is governed by complex interference patterns. [Learn more ...](Demos.ConicalInter.SpinBoson)

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Thu, 23 Feb 2017 12:33:00 -0000

--- v24
+++ v25
@@ -39,6 +39,11 @@

 If harmonic potentials are added to the linear E x e Jahn-Teller system, the prototypical spin-boson Hamiltonian is obtained with quadratic potentials on the diagonals and linear terms on the off-diagonal entries of the (diabatic!) potential matrix (κ ≠ 0, ω ≠ 0, g=0). Due to the multiple passing of high energy wavepackets of the conical intersection, the quantum dynamics is governed by complex interference patterns. [Learn more ...](Demos.ConicalInter.SpinBoson)

+Spin-Boson system in 1D
+---------------------
+
+In a diabatic representation, a one-dimensional spin-boson Hamiltonian can be represented by two harmonic potentials displaced in position and energy and a constant coupling between the two states. In the adiabatic representation, the two potential energy curves exhibit an avoided crossing. The corresponding quantum dynamics becomes incresingly complicated as time goes by: Upon each passage of the avoided crossing, each wavepacket spwns another one, eventually leading to rather complicated interference patterns.  [Learn more ...](Demos.ConicalInter.OneDim)
+
 Geometric phase effects
 -----------------------

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 07 Dec 2016 15:04:08 -0000

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 07 Dec 2016 15:03:05 -0000

--- v22
+++ v23
@@ -8,7 +8,7 @@

 Diabatic potential energy matrix (Cartesian coordinates)

-[[img src=Diabatic.gif alt="V(x,y)=\frac{\omega{2}\begin{pmatrix}x^2+y^2&0\\0&x^2+y^2\end{pmatrix}+\kappa\begin{pmatrix}x&y\\y&-x\end{pmatrix}+\frac{g}{2}\begin{pmatrix}x^2-y^2&-2xy\\-2xy&y^2-x^2\end{pmatrix}"]]
+[[img src=Diabatic.gif alt="V(x,y)=\frac{\omega}{2}\begin{pmatrix}x^2+y^2&0\\0&x^2+y^2\end{pmatrix}+\kappa\begin{pmatrix}x&y\\y&-x\end{pmatrix}+\frac{g}{2}\begin{pmatrix}x^2-y^2&-2xy\\-2xy&y^2-x^2\end{pmatrix}"]]

 Adiabatic potentials (using polar coordinates ρ and φ)

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 07 Dec 2016 15:02:09 -0000

--- v21
+++ v22
@@ -8,7 +8,7 @@

 Diabatic potential energy matrix (Cartesian coordinates)

-[[img src=Diabatic.gif alt="V(x)=\frac{\omega{2}\begin{pmatrix}x^2+y^2&0\\0&x^2+y^2\end{pmatrix}+\kappa\begin{pmatrix}x&y\\y&-x\end{pmatrix}+\frac{g}{2}\begin{pmatrix}x^2-y^2&-2xy\\-2xy&y^2-x^2\end{pmatrix}"]]
+[[img src=Diabatic.gif alt="V(x,y)=\frac{\omega{2}\begin{pmatrix}x^2+y^2&0\\0&x^2+y^2\end{pmatrix}+\kappa\begin{pmatrix}x&y\\y&-x\end{pmatrix}+\frac{g}{2}\begin{pmatrix}x^2-y^2&-2xy\\-2xy&y^2-x^2\end{pmatrix}"]]

 Adiabatic potentials (using polar coordinates ρ and φ)

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 07 Dec 2016 08:30:04 -0000

--- v20
+++ v21
@@ -47,4 +47,4 @@
 Quadratic E x e Jahn-Teller system: Warped potentials
 -----------------------------------------------------

-If quadratic Jahn-Teller coupling (g ≠ 0) is added to the spin boson system, the circular symmetry of the system is broken. The corresponding (adiabatic) potential energy surfaces become deformed ("warped") and exhibit threefold rotational symmetry with three additional conical intersections. The lower potential surface exhibits an alternating sequence of three minima and three saddles. [Learn more ...](Demos.ConicalInter.Warping)
+If quadratic Jahn-Teller coupling (g ≠ 0) is added to the spin boson system, the circular symmetry of the system is broken. The corresponding (adiabatic) potential energy surfaces become deformed ("warped") and the ground state exhibits threefold rotational symmetry with three additional conical intersections. The lower potential surface exhibits an alternating sequence of three minima and three saddles. [Learn more ...](Demos.ConicalInter.Warping)

Demos.ConicalInter.Main modified by Burkhard Schmidt

Burkhard Schmidt — Wed, 07 Dec 2016 08:20:47 -0000