Theory¶
Resonances and Non-Hermitian Quantum Mechanics¶
Electronic resonances are metastable electronic states with finite lifetimes embedded in the ionization/detachment continuum. Common examples include temporary anions formed by electron attachment, and core-excited and core-ionized states which can undergo Auger decay or similar relaxation pathways. These states are not part of the usual \(L^2\) Hilbert space of square integrable functions, and instead belong to the continuous spectrum of the electronic Hamiltonian. Theoretical description of resonances is generally not possible by means of conventional bound-state quantum chemistry methods, and special techniques are required to obtain accurate energies and lifetimes.
Non-Hermitian quantum mechanics (NHQM) techniques provide an attractive approach that enables adaptation of existing quantum chemistry methodologies to treat metastable electronic states. In NHQM formalisms, a resonance appears as a single square-integrable eigenstate of a non-Hermitian Hamiltonian, associated with a with a complex eigenvalue:
\(E=E_{res}-i\Gamma/2\).
The real part of the energy \((E_{res})\) is the resonance position. The imaginary part \((\Gamma/2)\) is the half-width, which is inversely proportional to the lifetime of the state [Reinhardt1982].
Complex Absorbing Potential¶
Complex absorbing potentials (CAPs) are imaginary potentials added to the Hamiltonian, and they are routinely used for evaluation of resonance parameters. In this context, CAPs transform a resonance into a single square integrable state, rendering it accessible by means of standard bound-state techniques. To this end, the electronic Hamiltonian is augmented with an imaginary potential:
\(H^{CAP}=H-i\eta W\)
where \(\eta\) is the CAP strength parameter, and W is a real potential which vanishes in the vicinity of the molecular system and grows with distance [Riss1993].
Since the CAP-augmented Hamiltonian depends on the strength of the CAP, a choice has to be made on the optimal value of \(\eta\) that provides best estimates of the resonance position and width. In a complete one-electron basis, the exact resonance position and width are obtained in the limit of an infinitesimally weak CAP \((\eta \rightarrow 0^+)\). In practice when finite bases are used, an optimal CAP strength \(\eta_{opt}\) is found by locating a stationary point on the eigenvalue trajectory E(\(\eta\)). A commonly used criterion is the minimum of the logarithmic velocity (\(|\eta\frac{dE}{d\eta}|\rightarrow min\)) [Riss1993].
Perturbative or “Projected” CAP¶
There are multiple strategies for how to incorporate CAPs into an electronic structure calculation. The most straightforward implementation is to engage the one-electron CAP term starting at the lowest level of theory (e.g. Hartree-Fock). While conceptually simple, this requires modification of electronic structure routines to handle the complex objects. Additionally, this approach requires a unique calculation for each \(\eta\) along the eigenvalue trajectory, which can become prohibitively expensive for larger systems or dynamical simulation.
An efficient alternative is to treat the CAP as a first order perturbation, considering only a small subset of the eigenstates of the real Hamiltonian [Sommerfeld2001]. In this case, the CAP will be introduced in the basis of the reduced subset of states:
\(W_{uv}=\langle u | W | v \rangle\)
where \(u\) and \(v\) are eigenstates of the real Hamiltonian. Since the CAP is a one-particle operator, these expressions can easily be evaluated using the CAP matrix in atomic orbital basis evaluated separately, the one-electron reduced density matrices (\(\rho\)) for each state, and the set of transition density matrices (\(\gamma\)) between each pair of states that are obtained from the bound-state calculation.
Once CAP matrix is evaluated the CAP-augmented Hamiltonian is constructed as follows:
\(H^{CAP}=H_0-i\eta W\)
where \(H_0\) is an appropriate zeroth order Hamiltonian obtained from the electronic structure calculation, and \(W\) is the CAP represented in the subspace. Diagonalization of this CAP-augmented Hamiltonian yields \(\eta\)-depdendent eigenvalues that are used to extract resonance position and width. Importantly, as only a small number of states in considered (typically less than 30), finding the eigenvalues of the CAP-augmented Hamiltonian has negligible cost in comparison to the bound-state electronic structure calculation required to get the initial set of states (u,v,..). Thus, although this perturbative or projected approach introduces another parameter (number of eigenstates), the overall cost is essentially reduced to that of a single electronic structure calculation.
With the zeroth order Hamiltonian and the CAP matrix, eigenvalue trajectories can be generated by means of simple external scripts, and estimates of resonances positions and widths can be obtained from analysis of the trajectories.
References¶
- Riss1993(1,2)
Riss, U. V.; Meyer, H. D. Calculation of Resonance Energies and Widths Using the Complex Absorbing Potential Method. J. Phys. B At. Mol. Opt. Phys. 1993, 26 (23), 4503–4535.
- Sommerfeld2001
Sommerfeld, T.; Santra, R. Efficient Method to Perform CAP/CI Calculations for Temporary Anions. Int. J. Quantum Chem. 2001, 82 (5), 218–226.
- Reinhardt1982
Reinhardt, W. P. Complex Coordinates in the Theory of Atomic and Molecular Structure and Dynamics. Annu. Rev. Phys. Chem. 1982, 33 (1), 223–255.