Algebraic modifications to second quantization for non‐Hermitian complex scaled hamiltonians with application to a quadratically convergent multiconfigurational self‐consistent field method

Abstract

ABSTRACT:The algebraic structure for creation and annihilation operators defined on orthogonal orbitals is generalized to permit easy development of bound-state techniques involving the use of non-Hermitian Hamiltonians arising from the use of complex-scaling or complex-absorbing potentials in the treatment of electron scattering resonances. These extensions are made possible by an orthogonal transformation of complex biorthogonal orbitals and states as opposed to the customary unitary transformation of real orthogonal orbitals and states and preserve all other formal and numerical simplicities of existing bound-state methods. The ease of application is demonstrated by deriving the modified equations for implementation of a quadratically convergent multiconfigurational self-consistent field (MCSCF) method for complexscaled Hamiltonians but the generalizations are equally applicable for the extension of other techniques such as single and multireference coupled cluster (CC) and many-body perturbation theory (MBPT) methods for their use in the treatment of resonances. This extends the domain of applicability of MCSCF, CC, MBPT, and methods based on MCSCF states to an accurate treatment of resonances while still using L 2 real basis sets. Modification of all other bound-state methods and codes should be similarly straightforward.

Complex-scaled multireference configuration-interaction method to study Be and Be-like cations' (B, C, N, O, Mg) Auger resonances1s2s22p1Zhang1, Yeager2 2012Phys. Rev. ASelf Cite211110View full textAdd to dashboardCite

“…The quadratically convergent complex-scaled multiconfigurational self-consistent field has been introduced in detail by Yeager et al [14]. After rotating the Hamiltonian into complex space, the complex-scaled Hamiltonian is not Hermitian; it is complex symmetric [15].…”

“…After rotating the Hamiltonian into complex space, the complex-scaled Hamiltonian is not Hermitian; it is complex symmetric [15]. Modified second quantization has been developed for complex-scaled non-Hermitian Hamiltonians [14,16], since the wave function |ψ m is complex conjugate biorthogonal (CCBON) where ψ * i |ψ j = δ ij ( * means complex conjugate). Creation operators are introduced as a T rather than a † and the usual anticommutation relations for creation and annihilation operators are kept by changing " †" into "T .…”

“…The complex scaled nonrelativistic electronic Hamiltonian for atoms is written as [14] (in atomic units)…”

“…F and G are the energy gradient vector and Hessian matrix, respectively, which have been explicitly defined in Refs. [8,9,14]. The complex variational principle [17][18][19] ensures that the search for the stationary point may be performed in the same way as MCSCF by setting the first derivative of Q(X)with respect to the step-length vector to zero, which leads to the multidimensional Newton-Raphson equation [8][9][10]14]:…”

Obtaining positions and widths of scattering resonances from a complex multiconfigurational self‐consistent field state using the M1 method

“…This prompted a modification 14, 34, 39 of the second quantization algebra 40 by introducing new sets of creation { a } and annihilation operators { a p } suitable for biorthonormal spin orbital bases, {φ p }. The antisymmetry of the spin orbitals with respect to exchange of any two electrons is maintained by the anticommutation relations of the creation and the annihilation operators.…”