--- title: "What changes when the electronic basis moves" description: "Derive derivative coupling from nuclear kinetic energy and compare adiabatic and diabatic representations with a two-state rotation." canonical_url: "https://sandboxphysics.com/stories/nonadiabatic-dynamics/moving-electronic-basis/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/nonadiabatic-dynamics/moving-electronic-basis/" last_modified: "2026-10-28" --- [Nonadiabatic molecular dynamics](https://sandboxphysics.com/stories/nonadiabatic-dynamics/) L2 · Chapter 2 # What changes when the electronic basis moves Sandbox Physics · 2026-10-28 As nuclei move, the electronic basis itself changes. Differentiating the molecular wavefunction therefore acts on both nuclear amplitudes and electronic states. These extra terms create derivative coupling. A basis change can move the coupling into the potential matrix without changing a fully retained prediction. ## Before you begin Read chapter one. You need complex amplitudes, matrix multiplication and the product rule. An electronic inner product integrates over electron coordinates at fixed nuclear position. We derive the equation for one nuclear coordinate. [Review the preceding chapter](https://sandboxphysics.com/stories/nonadiabatic-dynamics/potential-energy-surfaces/) - Obtain first- and second-derivative couplings from the product rule. - Explain rapid electronic mixing near a small gap. - Distinguish changing representation from dropping a coupling term. ## Expand the full state in a local electronic basis At each nuclear position $R$, choose orthonormal electronic eigenstates $\phi_n(r;R)$. Let $M$ be the nuclear-coordinate mass. Nuclear kinetic energy is $-\hbar^2\partial_R^2/(2M)$. Expand the molecular state with nuclear amplitudes $\chi_n(R,t)$. The nuclear derivative acts on both factors. Applying the product rule twice produces three terms. Multiplying by a second electronic state and integrating electron coordinates projects the molecular equation into coupled nuclear channels. $$ \Psi(r,R,t)=\sum_n\chi_n(R,t)\phi_n(r;R) $$ Electronic basis states depend on nuclear position; their coefficients depend on position and time. $$ \partial_R^2(\chi_n\phi_n)=\chi_n''\phi_n+2\chi_n'\phi_n'+\chi_n\phi_n'' $$ The two terms containing derivatives of the basis supply derivative coupling. Reading: Curchod · John C. Tully ## Nuclear motion meets the changing electronic basis Define $d_{mn}=\langle\phi_m|\partial_R\phi_n\rangle$ and $\tau_{mn}=\langle\phi_m|\partial_R^2\phi_n\rangle$. These inner products integrate electron coordinates only. Potential energy is diagonal in an adiabatic basis; nuclear kinetic energy mixes its channels. Along a classical nuclear path $R(t)$, the basis changes at a rate $\dot R\,d_{mn}$. Faster motion through the same spatial variation increases the rate at which the electronic state must adjust. $$ i\hbar\dot\chi_m=E_m\chi_m-\frac{\hbar^2}{2M}\sum_n\left(\delta_{mn}\partial_R^2+2d_{mn}\partial_R+\tau_{mn}\right)\chi_n $$ This equation retains all derivative terms within the selected electronic subspace. Truncating that subspace requires a separate convergence check. ## A rotation shows where the coupling goes Write a real symmetric two-state potential with common energy $s(R)$, diagonal difference $2z(R)$ and off-diagonal coupling $c(R)$. We call this smooth fixed basis diabatic. Nonadiabatic dynamics describes a process; diabatic describes a representation. A position-dependent rotation through angle $\theta(R)$ diagonalizes this matrix. It also transforms nuclear kinetic energy. The new derivative coupling contains the spatial derivative of the mixing angle. Dropping those terms after diagonalizing the potential changes the approximation. $$ V=sI+\begin{pmatrix}z&c\\c&-z\end{pmatrix},\quad E_\pm=s\pm\sqrt{z^2+c^2},\quad 2\theta=\operatorname{atan2}(c,z) $$ A continuous angle branch and consistent eigenvector signs must accompany this local expression. ## Worked example of a narrow coupling peak Near the crossing, set $z=aR$ and $c=\Delta>0$. Here $a$ has units of energy per length. Differentiation gives $\theta'=-a\Delta/[2(a^2R^2+\Delta^2)]$. The central magnitude is $|a|/(2\Delta)$. Halving the coupling doubles the derivative peak and halves its characteristic spatial width. The nucleus crosses a sharper change of electronic character. At zero coupling the eigenbasis is undefined at the degeneracy, so the divergent expression requires a different numerical treatment. ## A constant rotation supplies a small check At one position let the old coefficients be $(1,0)^{\mathsf T}$. Rotate the electronic basis through half a right angle. The new coefficients are $(1,1)^{\mathsf T}/\sqrt2$, with signs set by the rotation convention. Each new channel has probability one half and total probability remains one. To measure the original first state, rotate its projector too. Reading only the first new component instead would mistake a new coordinate axis for population transfer. This is why a channel probability needs a representation label. ## Test that representation has not changed the prediction For a unitary rotation $U(R)$, nuclear coefficients obey $\boldsymbol\chi_{\mathrm d}=U\boldsymbol\chi_{\mathrm a}$. The total density is unchanged point by point. Channel probabilities must be projected onto the same physical electronic states before comparison. A constant unitary rotation provides a simple solver check. A spatially changing rotation also transforms kinetic energy and requires derivative terms. Arbitrary sign flips of local eigenvectors must cancel in correctly transformed overlaps and probabilities. ## Try the prediction ### [Narrow the region of electronic mixing](https://sandboxphysics.com/labs/nonadiabatic-avoided-crossing/) 1. Keep incoming momentum fixed and predict how decreasing Electronic mixing changes the minimum gap. 2. Change only the mixing and compare the surfaces and outgoing branches. 3. Compare the sharper change of adiabatic character with the observed diabatic transfer, and explain the trend using the decoupled limit. ## Exercises and explanations 1. Why do adiabatic equations remain coupled after diagonalizing the potential? Read the explanation Nuclear kinetic energy differentiates the position-dependent basis. Cross derivatives and second derivatives of electronic states couple the channels. 2. Total density changes after a constant unitary rotation. What should you inspect first? Read the explanation Check that the potential, initial amplitudes and measurement projectors were all rotated consistently and that inner products were evaluated correctly. A constant unitary transformation preserves total density exactly. [Previous: How electrons create potential-energy surfaces](https://sandboxphysics.com/stories/nonadiabatic-dynamics/potential-energy-surfaces/) [Next: How much probability changes channel in one passage](https://sandboxphysics.com/stories/nonadiabatic-dynamics/landau-zener-passage/) ## Sources and calculation scope 1. [John C. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys. 93, 1061–1071 (1990)](https://doi.org/10.1063/1.459170) Published primary method paper. Sections II–III define fewest switches and its one-dimensional test models. Our three-method comparison uses the simple avoided-crossing Hamiltonian; the nuclear packet, finite windows and numerical settings are stated with each saved run. 2. [Curchod, Agostini & Gross, An Exact Factorization Perspective on Quantum Interferences in Nonadiabatic Dynamics, arXiv:1604.02351v2](https://arxiv.org/html/1604.02351v2) Fixed 30 June 2016 author version; published in J. Chem. Phys. 145, 034103 (2016), DOI 10.1063/1.4958637. Reading scope: equations (5)–(6), (14)–(18), and Figures 3–7. Its Shin–Metiu electron–ion calculation differs from the Tully and conical teaching models.