Bra State Research Articles

Time-dependent vibrational coupled cluster theory: Theory and implementation at the two-mode coupling level.

Equations are derived for the time evolution of time-dependent vibrational coupled cluster (TDVCC) wave functions covering both the TDVCC ket state and the associated so-called Λ bra state. The equations are implemented in the special case of both the Hamiltonian and the cluster operator containing at most two-mode coupling terms. The nontrivial behavior of the evolution of norm, energy, and expectation values due to the nonunitary time-evolution of the nonvariational TDVCC theory is analyzed theoretically and confirmed in numerical experiments that also include time-dependent Hamiltonians. In the spirit of time-independent size-consistency analysis, the separability of both the coupled cluster and Λ states for noninteracting systems is studied. While the coupled cluster state clearly has the correct behavior, the behavior of the Λ state is more intricate, and the consequence for different properties is shown theoretically and numerically. Overall, the numerical experiments show that TDVCC in incomplete expansions gives higher accuracy than a standard linear variational wave function parameterization with the same number of independent parameters, while equivalent results are obtained for complete expansions. The efficiency of the methodology is illustrated in computations on polycyclic aromatic hydrocarbons with up to 156 modes.

Parton shower evolution with subleading color

Abstract Parton shower Monte Carlo event generators in which the shower evolves from hard splittings to soft splittings generally use the leading color approximation, which is the leading term in an expansion in powers of ${{1} \left/ {{N_c^2}} \right.}$ , where N c = 3 is the number of colors. We introduce a more general approximation, the LC + approximation, that includes some of the color suppressed contributions. There is a cost: each generated event comes with a weight. There is a benefit: at each splitting the leading soft × collinear singularity and the leading collinear singularity are treated exactly with respect to color. In addition, an LC + shower can start from a state of the color density matrix in which the bra state color and the ket state color do not match.

Nontrivial Eigenvalues of the Liouvillian of an Open Quantum System

We present methods of finding complex eigenvalues of the Liouvillian of an open quantum system. The goal is to find eigenvalues that cannot be predicted from the eigenvalues of the corresponding Hamiltonian. Our model is a T-type quantum dot with an infinitely long lead. We suggest the existence of the non-trivial eigenvalues of the Liouvillian in two ways: one way is to show that the original problem reduces to the problem of a two-particle Hamiltonian with a two-body interaction and the other way is to show that diagram expansion of the Green's function has correlation between the bra state and the ket state. We also introduce the integral equations equivalent to the original eigenvalue problem.

Duality in Interacting Particle Systems and Boson Representation

In the context of Markov processes, we show a new scheme to derive dual processes and a duality function based on a boson representation. This scheme is applicable to a case in which a generator is expressed by boson creation and annihilation operators. For some stochastic processes, duality relations have been known, which connect continuous time Markov processes with discrete state space and those with continuous state space. We clarify that using a generating function approach and the Doi-Peliti method, a birth-death process (or discrete random walk model) is naturally connected to a differential equation with continuous variables, which would be interpreted as a dual Markov process. The key point in the derivation is to use bosonic coherent states as a bra state, instead of a conventional projection state. As examples, we apply the scheme to a simple birth-coagulation process and a Brownian momentum process. The generator of the Brownian momentum process is written by elements of the SU(1,1) algebra, and using a boson realization of SU(1,1) we show that the same scheme is available.

Generalizing the bra state in the symmetry-adapted cluster singles and doubles method and the second-order perturbation correction

We present a procedure to overcome the quasi-degeneracy problem in the CCSD and SACSD methods. A multi-configuration bra state in the SACSD equations (MRbra-SACSD equation) together with a second-order perturbation correction gives quantitatively correct potential curves for bond dissociations. The benchmark results for HF, F 2, and H 2O show that the perturbative triple correction reduces the deviations from Full-CI energies to within 3 mHartree (1.9 kcal/mol). Within the benchmark calculations, the results of the present method are comparable with the recent post-CCSD corrections, the CR-CCSD(T) and CCSD(2) T methods, and are slightly less accurate than the CR-CC(2,3) method.

Analytic and contour representations in the unit disk based on SU(1,1) coherent states

A contour representation in the unit disk based on SU(1,1) coherent states is introduced. The scalar product is given by a contour integral. The regions of convergence of the functions representing ket and bra states are studied. An analytic representation in the unit disk is also considered, where the scalar product is represented by an integral over the unit disk, with the Lobachevsky measure. Various relations which connect these analytic functions with other phase-space quantities are derived.

A Theoretical Distinction Between Time-Resolved Resonance Raman and Resonance Fluorescence

Based on the time-dependent theory, an analysis of the distinction between resonance Raman (RR) and resonance fluorescence (RF) with pulse excitation was presented. The real population of the intermediate state gives two optical components-the independent time evolution of intermediate ket and bra states generates RR while RF originates from the phase coherent between ket and bra states. In cw limit, the transition probability of spontaneous emission with pulse excitation can be reduced to the classical theory.

Separability properties of reduced and effective density matrices in the equation-of-motion coupled cluster method

Certain aspects of final state descriptions provided by the equation-of-motion coupled cluster (EOM-CC) method are analyzed, particularly the asymptotic separability of density matrices. Specific attention is focused on a supermolecule system that consists of two fragments, one of which is in an excited electronic state. For this example, the reduced n-particle density matrix [ρ(n)] associated with EOM-CC theory exhibits unphysical long range behavior due to a lack of bra state multiplicative separability. More satisfactory is the effective density [D(n)] which includes the first order response of ground state correlation. In particular, local components of D(n) reduce to the corresponding monomer quantities in the limit of infinite separation. This feature guarantees that observables calculated by contracting D(n) with local operators are size consistent. However, the situation with respect to physical phenomena associated with nonlocal operators is less clear, as multicenter matrix elements of both ρ(n) and D(n) behave incorrectly for n≥2 in the separated limit. The latter consideration should not be misconstrued as an inherent failure of the EOM-CC approximation, as problems associated with nonlocal parts of the density are already present in the normal coupled cluster treatment of the ground state.

Diagrammatic analysis and application of the coupled cluster response approach to ground‐state expectation values

AbstractThe response approach to couple cluster (CC) expectation values devised by Monkhorst and the normal coupled cluster method (NCCM) discussed by Arponen and Bishop are considered from a diagrammatic point of view. The perturbative diagrammatic content of the operator that parametrizes the NCCM bra state is discussed in detail. The method is applied to the calculation of the one‐particle reduced density matrix of the Be atom and the HF molecule for different values of the internuclear distance. Various contributions to the total energy obtained in the NCCM framework are compared to results from accurate multireference configuration interaction calculations. Such a comparison is shown to provide a much more stringent test on the accuracy of the CC formalism than will a comparison of total energy alone. © 1993 John Wiley & Sons, Inc.

Configuration interaction in a basis of biorthogonal states

This article develops techniques for configuration interaction in a nonorthogonal basis. Hamiltonian operator matrix elements such as 〈φ̃u‖Ĥ‖φv〉 are easily evaluated when biorthogonal orbitals are used for the bra and ket configuration states. The ket configuration state ‖φv〉 is constructed from a basis of orbitals {Xi} and the bra configuration state 〈φ̃u‖ employs the dual orbitals {X̃i}. The dual basis is defined by contraction of the original basis with the inverse overlap matrix: X̃i(r)= ∑jXj(r)S−1ji. A configuration interaction wave function ‖ψ〉 then corresponds to a stationary point of the modified energy functional E′=〈ψ̃‖Ĥ‖ψ〉/〈ψ̃‖ψ〉. This energy functional becomes exact as either ‖ψ〉 or ‖ψ̃〉 approaches an exact eigenstate of the Schrödinger equation. The wave function ‖ψ̃〉 is expanded with respect to the dual basis, whereas the original configuration state basis is employed for expansion of ‖ψ〉. The energy E′ satisfies the variation theorem if and only if the dual basis is related to the original basis by a linear transformation. Such a transformation exists when a full CI is done with respect to the manifold defined by nonorthogonal active orbitals. However, one can retain nonorthogonality among those orbitals which are doubly occupied in all configurations without invalidating the variational stability. Configuration interaction in a nonorthogonal basis can be efficiently done in direct CI mode. Modification of the second-quantization formalism shows that the structure constants of the direct CI are identical with those that would pertain to a nonorthogonal basis. The dual basis that is used for the bra states modifies the integral transformation, but otherwise has no effect.