Editing Geometric phase (section)

=== Exposure in molecular adiabatic potential surface intersections ===

There are several ways to compute the geometric phase in molecules within the [[Born–Oppenheimer]] framework. One way is through the "non-adiabatic coupling <math>M \times M</math> matrix" defined by
<math display="block">
\tau_{ij}^\mu = \langle \psi_i | \partial^\mu \psi_j \rangle,
</math>
where <math>\psi_i</math> is the adiabatic electronic wave function, depending on the nuclear parameters <math>R_\mu</math>. The nonadiabatic coupling can be used to define a loop integral, analogous to a [[Wilson loop]] (1974) in field theory, developed independently for molecular framework by M.&nbsp;Baer (1975, 1980, 2000). Given a closed loop <math>\Gamma</math>, parameterized by  <math>R_\mu(t),</math> where <math>t \in [0, 1]</math> is a parameter, and <math>R_\mu(t + 1) = R_\mu(t)</math>. The ''D''-matrix is given by
<math display="block">
D[\Gamma] = \hat{P} e^{\oint_\Gamma \tau^\mu \,dR_\mu}</math>
(here <math>\hat{P}</math> is a path-ordering symbol). It can be shown that once <math>M</math> is large enough (i.e. a sufficient number of electronic states is considered), this matrix is diagonal, with the diagonal elements equal to <math>e^{i\beta_j},</math> where <math>\beta_j</math> are the geometric phases associated with the loop for the <math>j</math>-th adiabatic electronic state.

For time-reversal symmetrical electronic Hamiltonians the geometric phase reflects the number of conical intersections encircled by the loop. More accurately,
<math display="block">
e^{i\beta_j} = (-1)^{N_j},
</math>
where <math>N_j</math> is the number of conical intersections involving the adiabatic state <math>\psi_j</math> encircled by the loop <math>\Gamma.</math>

An alternative to the ''D''-matrix approach would be a direct calculation of the Pancharatnam phase. This is especially useful if one is interested only in the geometric phases of a single adiabatic state. In this approach, one takes a  number <math>N + 1</math> of points <math>(n = 0, \dots, N)</math> along the loop <math>R(t_n)</math> with <math>t_0 = 0</math> and <math>t_N = 1,</math> then using only the ''j''-th adiabatic states <math>\psi_j[R(t_n)]</math> computes the Pancharatnam product of overlaps:
<math display="block">
I_j(\Gamma, N) = \prod\limits_{n=0}^{N-1} \langle \psi_j[R(t_n)] | \psi_j[R(t_{n+1})] \rangle.
</math>

In the limit <math>N \to \infty </math> one has (see Ryb & Baer 2004 for explanation and some applications)
<math display="block">
I_j(\Gamma, N) \to e^{i\beta_j}.
</math>