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The parallel-transported (quasi)-diabatic basis.

Robert Littlejohn1, Jonathan Rawlinson2, Joseph Subotnik3

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This study introduces a parallel-transported basis for simplifying calculations near electronic degeneracies. This method facilitates Taylor series expansions and efficiently computes derivative couplings, crucial for quantum mechanics.

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Area of Science:

  • Theoretical Chemistry
  • Quantum Mechanics
  • Computational Chemistry

Background:

  • Analytic treatments of the nuclear Schrödinger equation near degeneracies are complex.
  • Existing methods struggle with singularities and energy denominators in expansions.
  • The electronic Hamiltonian can exhibit degeneracies, forming seams in potential energy surfaces.

Purpose of the Study:

  • To introduce and explore the utility of a parallel-transported basis for quantum mechanical calculations.
  • To demonstrate the advantages of this basis for analytic treatments near electronic degeneracies.
  • To establish an efficient method for calculating Taylor series expansions and derivative couplings.

Main Methods:

  • Utilizing parallel transport to construct a diabatic basis.
  • Expanding both the parallel-transported and singular-value bases in power series.
  • Employing a projection operator for Taylor series expansions, avoiding energy denominators.

Main Results:

  • The parallel-transported basis simplifies Taylor series expansions near points and seams.
  • Derivative couplings show a direct relationship with curvature in this basis.
  • The parallel-transported basis agrees with the singular-value basis up to second-order terms.

Conclusions:

  • The parallel-transported basis, analogous to Poincaré gauge, offers an efficient method for calculations.
  • This approach, combined with Mead's formula, streamlines the computation of basis states and derivative couplings.
  • The method is applicable even when fine structure effects are included in the electronic Hamiltonian.