--- title: "How two surfaces form a conical intersection" description: "Derive the two conditions for a real two-state degeneracy, construct a local cone, and examine how the incident path affects branching." canonical_url: "https://sandboxphysics.com/stories/nonadiabatic-dynamics/conical-intersections/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/nonadiabatic-dynamics/conical-intersections/" last_modified: "2026-10-28" --- [Nonadiabatic molecular dynamics](https://sandboxphysics.com/stories/nonadiabatic-dynamics/) L3 · Chapter 5 # How two surfaces form a conical intersection Sandbox Physics · 2026-10-28 Two electronic energy surfaces can meet exactly at a nuclear configuration. One direction reduces their energy difference; an independent direction changes their mixing. Plot both directions and the contact looks like two opposing cones. ## Before you begin Read chapters two and four. You need two-dimensional coordinates and eigenvalues of a two-state matrix. Both axes describe nuclear configuration changes; they need not be Cartesian displacements of one atom. [Review the preceding chapter](https://sandboxphysics.com/stories/nonadiabatic-dynamics/three-dynamics/) - Identify the two conditions for a real two-state degeneracy. - Derive the local cone and its energy gap. - Separate packet branching, finite-window probabilities and geometric phase. ## A generic single coordinate cannot satisfy both conditions A real symmetric two-state matrix has common energy $s$, half-difference $z$ and coupling $c$. Degeneracy requires $z=0$ and $c=0$. A generic one-coordinate scan cannot independently tune both, so a finite coupling leaves an avoided crossing. Introduce a tuning coordinate $x$ and mixing coordinate $y$. Locally set $z=ax$ and $c=by$ with nonzero energy gradients $a$ and $b$. The degeneracy lies at the origin. $$ V(x,y)=s(x,y)I+\begin{pmatrix}ax&by\\by&-ax\end{pmatrix},\qquad E_\pm=s\pm\sqrt{a^2x^2+b^2y^2} $$ This is a local real two-state model without spin–orbit coupling. Complex couplings, additional states and symmetry can change the degeneracy structure. $$ \Delta E=2\sqrt{a^2x^2+b^2y^2} $$ The gap grows linearly with distance from contact along either independent direction. ## Both coordinates matter to a passing packet Head-on and off-center packets encounter different gaps and changes of electronic character. A packet has transverse width, so even a center aimed at the degeneracy carries amplitude past both sides. A single central trajectory loses that structure. The existing conical-intersection experiment evolves a two-component wavefunction in a dimensionless linear vibronic-coupling model with common potential $s=\omega^2(x^2+y^2)/2$. The fixed dimensionless confinement frequency is 0.65; it controls the strength of the common harmonic potential. Surface color and readouts must refer consistently to either fixed channels or adiabatic projections. ## Worked example of an offset straight path Prescribe $x=vt$ and $y=b_0$ with $v>0$. The half-difference is $avt$ and coupling is $bb_0$. The minimum gap is $2|bb_0|$, so doubling the offset doubles the gap. Under the additional assumptions of one uniform, asymptotically long passage, the Landau–Zener parameter is $\Gamma=b^2b_0^2/(2\hbar|av|)$. The full diabatic difference changes at rate $2av$. Doubling offset therefore quadruples the exponent’s magnitude. This estimates a prescribed local path. A two-dimensional nuclear packet spreads and bends, with different components sampling different couplings. Full propagation is needed for its branching probabilities. Reading: C. Zener ## Electronic population depends on the projector A diabatic channel probability integrates the squared magnitude of a fixed component. Lower-adiabatic population first applies the local projector $P_-(x,y)$. They answer different questions and may have different values. Total density adds both component densities and is preserved by a consistent unitary change of basis. At the degeneracy the individual eigenstate is not unique. Its zero-measure contribution vanishes in continuous integrals; a finite grid needs a separately checked convention. $$ P_-(t)=\int d x\,d y\;\boldsymbol\chi^\dagger(x,y,t)P_-(x,y)\boldsymbol\chi(x,y,t) $$ The left side denotes integrated lower-surface population; the matrix inside the integral is the local projector. ## After branching, ask what happens to phase Separate density lobes show nuclear branching. If amplitudes meet again, their relative phase also controls the density. Paths around a conical intersection can recombine destructively. To identify geometric phase, compare electronic states around a closed circuit or controlled interference with the phase retained and removed. A change in one instantaneous surface population does not isolate that effect. The next chapter specifies paths and recombination so the phase can be tested separately. Reading: M. V. Berry ## Try the prediction ### [Change the incident offset](https://sandboxphysics.com/labs/molecular-conical-intersection/) 1. Predict which preparation samples the smallest gaps: head-on or off-center. 2. Change only the offset and compare density and lower-surface population at the same time. 3. Use the no-mixing control and explain which electronic transfer disappears while nuclear evolution continues. ## Exercises and explanations 1. Why does a generic real two-state scan with one coordinate avoid crossing? Read the explanation Degeneracy requires both the diagonal difference and real off-diagonal coupling to vanish. These are generally two independent conditions. 2. Do different diabatic and adiabatic populations imply a failure of probability conservation? Read the explanation They use different projectors. Check the sum over each complete set against total probability; individual channels should be projected onto the same electronic states before comparison. [Previous: Three dynamics methods for the same preparation](https://sandboxphysics.com/stories/nonadiabatic-dynamics/three-dynamics/) [Next: Why paths around a conical intersection can cancel](https://sandboxphysics.com/stories/nonadiabatic-dynamics/geometric-phase-interference/) ## Sources and calculation scope 1. [M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392, 45–57 (1984)](https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry120.pdf) Published primary derivation: equations (5)–(6) separate dynamical and geometrical phase; section 3 treats a degeneracy. The path experiment is a controlled two-state model, with the reduction and phase comparison specified on its page. 2. [C. Zener, Non-adiabatic crossing of energy levels, Proc. R. Soc. Lond. A 137, 696–702 (1932)](https://doi.org/10.1098/rspa.1932.0165) Published original result for a linear-in-time diabatic gap and constant coupling. The course defines the slope as the slope of the full diabatic energy difference, avoiding a factor-of-two ambiguity.