<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

The Born-Oppenheimer approximation is one of the most important simplifications to the many-body Schrodinger equation. By neglecting nuclear motion, the nuclear and electronic motion can be separated, in what is known as an adiabatic system. In this adiabatic regime, nuclear motion occurs incrementally such that the system remains an energy eigenfunction of the instantaneous Hamiltonian at all times. The Born-Oppenheimer approximation leads to the typical paradigm of electronic, rotational, and vibrational degrees of freedom, which can be calculated independently. The limitations of the Born-Oppenheimer regime occur when nuclear motion is coupled to electronic motion, in what is known as vibronic coupling. This diabatic regime commonly occurs in photochemistry or chemical reactions, where nuclear motion becomes significant enough for vibronic coupling to come into play. For every adiabatic state, a potential energy surface (PES) can be plotted. As shown in the theory section below, different excited states potential energy surfaces are often degenerate at a single point, which forms a cone-like shape where the two surfaces intersect. This is the conical intersection, the 0th dimensional space where degeneracy is possible. Conical intersections are key to understanding transitions between states, particularly in excited state dynamics such as that which occurs in photochemistry. For instance in fluorescence, a transition occurs from a singlet excited state S1 to a singlet ground state S0, which may occur as a transition down the seam of a conical intersections.

Conical intersections introduce what is known as a Berry phase (more generally, the geometric phase) to the wavefunction, a measurable phase shift in the electronic wavefunction that is induced by cyclical processes. In an adiabatic system where the wavefunction is parametrized by the nuclear coordinates, motion through parameter space generates a phase factor. The sign of the nuclear wavefunction changes with the opposite of the electronic wavefunction in order to preserve the sign of the total wavefunction.

A closely related phenomenon to that of the conical intersection is that of Dirac cones. This occurs in the solid state, where valence and conduction bands meet at Dirac points, forming the same cone-like structure as conical intersections. In k-space (the space containing the crystal momentum k in the plane wave basis), Dirac cones occur at the vertices of the Brillouin zone (the unit cell in k-space). Dirac cones produce similar phenomena to CIs such as Berry phase shifts, which contribute to phenomena such as the anomalous Hall effect and topological insulators.

Theory/Methodology:

Born-Oppenheimer Approximation:

The total Hamiltonian can be written as

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Or in expanded form:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS1GLDYuUTQmUmlnaHRBbmdsZUJyYWNrZXQ7RidGL0YyL0Y2USV0cnVlRidGNy9GOkZMRjtGPUY/RkFGQ0ZGRi8vJTBmb250X3N0eWxlX25hbWVHUSVUZXh0RidGMg== = En 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The Born-Oppenheimer approximation assumes that the electronic and nuclear wavefunctions are separable

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Since the motion of the nuclei occurs on a much slower scale than the electronic motion, the kinetic energy term can be neglected.

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The electronic wavefunction now only has parametric dependence on the nuclear positions 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Potential Energy Surfaces

An important tool to visualize the energy of a molecule is to plot the energy as a function of nuclear coordinates. This is done under the Born-Oppenheimer approximation, where it is assumed the electrons instantly adapt to incrementally increasing nuclear coordinates. This adiabatic approximation yields potential energy surfaces, multi-dimensional functions of the nuclear coordinates.

Avoided Crossing:

The breakdown of the Born-Oppenheimer approximation can be described perturbatively. A diabatic perturbation is applied to the system, which can be analyzed with degenerate perturbation theory.

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Since 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

Solving for the eigenvalues now yields

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JJm1mcmFjR0YkNigtRiM2KC1JI21uR0YkNiRRIjFGJ0Y5Ri8vJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRjEvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjFGMi1GIzYlLUZlbjYkUSIyRidGOUYvRjIvJS5saW5ldGhpY2tuZXNzR0Znbi8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zoby8lKWJldmVsbGVkR0Y9LUkobWZlbmNlZEdGJDYkLUYjNi8tSSVtc3ViR0YkNiUtRiw2JkYuRi8vJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjItRiM2JC1GZW42JUZnbkZncEY5RjkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GNjYuRjhGZ3BGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GNjYuUSIrRidGZ3BGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmdxRmFxLUZjcDYlRmVwLUYjNiQtRmVuNiVGY29GZ3BGOUY5Rl5xRmFxRmNxRmFxLUZjcDYmLUYsNidRIlZGJ0YvRltvRmdwRjItRiM2JUZccUZbb0Y5Rl5xRltvLUY2Ni1GZXFGOUY7Rj5GQEZCRkRGRkZIRmZxRmhxLUZjcDYmLUYsNiZGY3JGL0Zbb0YyLUYjNiVGXXJGW29GOUZecUZbby8lK2V4ZWN1dGFibGVHRj1GOUY5RmFxLUY2Ni5RJSZwbTtGJ0ZncEY5RjtGPkZARkJGREZGRkhGZnFGaHFGYXEtRiM2JS1GNjYtUSFGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLUkmbXNxcnRHRiQ2Iy1GIzYoLUklbXN1cEdGJDYlLUZecDYlLUYjNi5GYnBGYXEtRjY2LlEqJnVtaW51czA7RidGZ3BGOUY7Rj5GQEZCRkRGRkZIRmZxRmhxRmFxRmlxRjVGZnJGNS1GY3A2JkZqckZkckZecUZbby1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRmZxRmhxRmhyRjlGZ3BGOUZfby8lMXN1cGVyc2NyaXB0c2hpZnRHRmBxRmNxLUZlbjYlUSI0RidGZ3BGOUZhcS1GXnQ2JS1GIzYoRmVzLUYjNiVGZXMtRl5wNiYtRiM2JS1GLDYmRmNyRi9GZ3BGMkZec0Y5RjkvJSVvcGVuR1EmJm1pZDtGJy8lJmNsb3NlR0ZfdkY5RmVzRi9GZ3BGMkZfb0ZcdUY5RjlGZXNGXnNGOQ==

For a crossing of states, the two solutions must be degenerate, 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= 0

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

The energy is a function of the molecular geometry, so the system of equation must be solved using the parameters corresponding to nuclear positions.

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

plot([sqrt(x^2+1),-sqrt(x^2+1), x, -x], x = -4..4, color = [red, blue, black, black], linestyle = [solid, solid, dash, dash], labels = [BondDistance, Energy])

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For a diatomic molecule, where only one nuclear coordinate exists, there is only one degree of freedom (the bond length) and this no solution to this system of equations exists. This results in what is known as an avoided crossing, where the potential energy surfaces of two states become very close, but do not actually cross. If a diabatic basis is used (see theory section) instead of an adiabatic basis, then the states do appear to cross, as indicated by the dashed black lines in the above figure. The idealized avoided crossing takes the form of a hyperbola (see figure 1). For polyatomic molecules, the multiple internuclear distances provide the degrees of freedom necessary for there exists a single solution to the equations. It is at this point at which the conical intersection occurs.

plot3d([x^2 + y^2, -x^2 - y^2], x = -10..10, title = "Conical Intersection", labels = [R[1], R[2], Energy]);

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Adiabatic Theorem and Geometric (Berry) phase:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==

The Born-Oppenheimer approximation leads to the wavefunction ass

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

The time-dependent Schr\303\266dinger equation is

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

Writing as an ansatz:

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 = 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

A simple example where the system depends on a singular nuclear coordinate will suffice to illustrate the adiabatic theorem. In the adiabatic regime, the system will be an eigenstate of the Hamiltonian at every time point, thus

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= 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

The system begins in the initial state

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LlEuJlZlcnRpY2FsQmFyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRi1JI21pR0YkNiZRJyYjOTM2O0YnLyUnaXRhbGljR0Y3Ri9GMi1JKG1mZW5jZWRHRiQ2JS1GIzYmLUZKNihRIlJGJy8lJWJvbGRHUSV0cnVlRicvRk5GWUYvL0YzUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GLDYwUSIoRidGV0YvL0YzRmluRmduL0Y2RllGOC9GO0ZZRjxGPkZARkIvRkVRLDAuMTY2NjY2N2VtRicvRkhGYW8tSSNtbkdGJDYlUSIwRidGL0YyRjJGL0YyLUYsNi5RNCZSaWdodEFuZ2xlQnJhY2tldDtGJ0YvRjJGXm9GOEZfb0Y8Rj5GQEZCL0ZFUSYwLjBlbUYnL0ZIRltwLUYsNi5RIn5GJ0YvRjJGNUY4RjpGPEY+RkBGQkZqb0ZccEYy

Plugging into the TDSE and projecting the eigenbasis

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=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

Using the product and chain rule to expand the time derivative

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

The adiabatic approximation means that the terms in the summation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJSZuZTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJuRidGL0YyLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVi8lK2V4ZWN1dGFibGVHRj1GOQ==can be ignored

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

Integrating this gives

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

Where

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmdhbW1hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEidEYnRj1GP0Y1RjVGNQ== = 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

The total wavefunction can be written as

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

More generally, for a multi-dimensional problem the Berry phase is

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEoJmdhbW1hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JC1GLzYlUSJtRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRi82JVEidEYnRj1GP0Y1RjVGNQ== = 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

Where the Berry connection is

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

Thus, the Berry phase can be written as a path integral independent of timeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==

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

Or alternatively,

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

Where LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFEoJk9tZWdhO0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3 is the Berry curvature, a second rank tensor.

The Berry phase is Gauge invariant, in other words, there is a freedom to construct the phase factor by multiplying LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JJW1yb3dHRiQ2JC1GLDYmUSJtRicvRjBRJXRydWVGJ0YyL0Y2USdpdGFsaWNGJ0Y1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= by some factor 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

The Berry phase is observed as a change in the sign of the wavefunction as it circles the conical intersection. A rotation around the CI in parameter space results in a path-dependent Berry phase. The exact value of this phase depends on the number of CIs enclosed by the path. An odd number of CIs enclosed results in an odd multiple of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSUmcGk7RicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRi8=, and vice versa for even numbers of enclosed CIs. If no CIs are surrounded, then the Berry phase is zero. The Chern theorem postulates thatLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkMtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGQy8lJmRlcHRoR0ZLLyUqbGluZWJyZWFrR1EobmV3bGluZUYnLUZHNiZGSUZMRk4vRlFRJWF1dG9GJ0YrRi8=

Diabatic Representations:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkNGLw==

Diabatic states are used to describe rapid processes where the adiabatic theorem does not hold. The coupling term in this case is the non-negligible nuclear kinetic energy term, which becomes relevant during rapid molecular motion. Diabatic states are found by diagonalizing the nuclear kinetic term in the Hamiltonian. This is achieved by a unitary transformation from the adiabatic basis to the diabatic:

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

In the diabatic basis, avoided crossings that do not intersect because of symmetry are seen to intersect, as shown above.