Generalized density functional theory for degenerate states

An extension of density functional theory is proposed for degenerate states. There are suitably selected basic variables beyond the subspace density. Generalized Kohn-Sham equations are derived. A direct method is proposed to ensure the fixed value of ensemble quantities. Then the Kohn-Sham equations are similar to the conventional Kohn-Sham equations. But the Kohn-Sham potential is different for different ensembles. A simple local expression is proposed for the correlation energy.     

The Landau-Zener formula

The Landau-Zener formula for the probability that a nonadiabatic transition has taken place is derived without solving directly the usual second-order differential equation. This is achieved in just a few steps by using contour integration.     

Novel spectroscopic effects in molecular transitions involving degenerate electronic states

We discuss the intricate Jahn—Teller-like and non-Jahn—Teller spectroscopic patterns which result via the recently revealed cross-quadratic terms involves modes of different symmetry which arise in the nuclear potential of degenerate electronic states of non-linear molecules.     

A quantum-classical treatment of non-adiabatic transitions

We have carried out molecular dynamics simulations of non-adiabatic processes with the help of a newly formulated potentially exact quantum-classical approach derived from a method proposed earlier [J. Chem. Phys. 118 (2003) 5302]. In this method, time-dependent Schroedinger equation is solved by representing Ψ on a moving Gauss–Hermite DVR grid, the motion of grid-centre being handled classically, but self consistently with the quantum evolution of the wavefunction. Electronic transitions are allowed anywhere in the configuration space among any number of coupled states. We have tested the method on three model problems proposed by J.C. Tully [J. Chem. Phys. 93 (1990) 1061]. These models are relevant to a wide range of gas-phase and condensed-phase phenomena occurring even at low energies. Excellent agreement of computed transition probabilities with corresponding quantum mechanical (DVR/FFT) results even in the deep quantum regime and its cost-efficiency (computational) are encouraging.     

Study of the non-adiabatic transitions with application to NO

An expression for the non-adiabatic transition probability is derived from the viewpoint of the non-stationary character of the adiabatic approximation. A numerical calculation has been made for the free NO molecule. The non-adiabatic transition probability for the transition (B ² =0)(a ⁴ =9) is estimated to be 10^–6 sec^–1 by using the wave functions proposed by Moser et al.

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Zusammenfassung Für die nicht adiabatische Übergangswahrscheinlichkeit wurde aus dem nicht-stationären Charakter der adiabatischen Näherung ein Ausdruck hergeleitet, der für den Fall des NO-Moleküls numerisch ausgewertet wurde. Dabei ergab sich unter Verwendung der Wellenfunktionen von Moser u. Mitarb. eine Wahrscheinlichkeit für den Übergang (B ² =0) (a ⁴ =9) von der Größenordnung von 10^–6 sec^–1.

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Résumé Une expression pour la probabilité de la transition non adiabatique est obtenue du point de vue du caractère non stationnaire de l'approximation adiabatique. Un calcul numérique a été effectué pour la molécule NO isolée. La probabilité de transition non adiabatique pour la transition (B ² =0)(a ⁴ =9) est évaluée à 10^–6 sec^–1 en utilisant les fonctions d'onde proposées par Moser et al.

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This work was supported by The Faculty Grant of Arizona State University.     

Radiationless transitions between excited singlet states of biacetyl

Emission processes from lower excited states S₁ (fluorescence) and T₁ (phosphorescence) have been studied in the gas and liquid phases when biacetyl is excited into the second singlet state S₂. (In agreement with Kasha's rule no fluorescence is observed from the S₂ state.) In the liquid phase, when biacetyl is excited into the singlet states S₁ and S₂, no difference is observed between these emission processes. This phenomenon certainly results from an efficient nonradiative transition between the second excited singlet state S₂ and the first excited state S₁ with practically no excess vibrational energy. The quantum yield of this transition is almost unity and does not depend on the nature of the solvent. In the gas phase no emission processes are observed when biacetyl is excited into the S₂ state at low pressure (less than 10 mm Hg). High pressure of inert gas is necessary in order to observe these processes. As for excitation into the S₁ state with vibrational energy, loss of vibrational energy through collisions occurs from the S₂ state. The quantum yield of the S₂ → S₁ transition by excitation at 290 nm is estimated around 0.5–0.6 at 6 atm of inert gas (ethane, ethylene, or carbon dioxide).     

Comparison of multichannel and two-state calculations for H-atom transfer between two nearly degenerate states

Multichannel and two-state calculations are made for a collinear H-atom transfer between two heavy masses. The particular reaction studied is on where the initial vibrational state of the reactants is nearly degenerate with a vibrational state of the products. The two sets of results for the reaction probability are in good agreement with each other.     

A limitation of two-state analysis for transitions between disordered and weakly ordered states

Background: The stability of the secondary structure of particular peptide regions is often used to investigate the involvement of the region in protein folding. When analysing the relatively small populations of associated states that are formed by weak interactions (i.e. those interactions that are comparable to thermal energies), it is common practice to characterise the associated state by a parameter that is measured when this state is highly occupied. The accuracy of this method, however, has not yet been determined.Results: Using as a model the vancomycin group of antibiotics, either forming dimers or binding to cell wall precursors, we have investigated the dependence of the limiting (i.e. fully associated) chemical shifts of two protons on the equilibrium constants for the formation of the fully associated states. The chemical shift shows a large variation with the equilibrium constant for the formation of the fully associated state.Conclusions: The results demonstrate, in two systems, that a parameter representing a fully associated state (chemical shift) varies greatly with the equilibrium constant for the formation of that associated state. The results have implications for two-state analyses of populations of protein fragments in which a parameter representing the fully associated state is taken to be independent of the equilibrium constant for its formation. Using two-state analysis to determine the population of associated states of protein fragments could result in an underestimation of the population of these associated states.     

Generalized simulated tempering for exploring strong phase transitions

An extension of the simulation tempering algorithm is proposed. It is shown to be particularly suited to the exploration of first-order phase transition systems characterized by the backbending or S-loop in the statistical temperature or a microcanonical caloric curve. A guided Markov process in an auxiliary parameter space systematically combines a set of parametrized Tsallis-weight ensemble simulations, which are targeted to transform unstable or metastable energy states of canonical ensembles into stable ones and smoothly join ordered and disordered phases across phase transition regions via a succession of unimodal energy distributions. The inverse mapping between the sampling weight and the effective temperature enables an optimal selection of relevant Tsallis-weight parameters. A semianalytic expression for the biasing weight in parameter space is adaptively updated "on the fly" during the simulation to achieve rapid convergence. Accelerated tunneling transitions with a comprehensive sampling for phase-coexistent states are explicitly demonstrated in systems subject to strong hysteresis including Potts and Ising spin models and a 147 atom Lennard-Jones cluster.     

A magnetic resonance study of non-adiabatic evolution of spin quantum states

We use a nuclear-spin gyroscope for demonstrating non-adiabatic rotational effects and the geometric phase of a polarized Xe nuclear-spin ensemble. We treat a spin-1/2 model system in a rotating magnetic field (with its inherent single-valued eigenstates) and a spin-3/2 system in a rotating electric field gradient (with a partially degenerate eigenstate structure) including (i) spin polarization by polarization transfer from optically pumped alkali atoms, (ii) spin-coherence excitation, and (iii) detection. The results are compared to those obtained by the concept of geometric phases, where Berry's phase emerges as the adiabatic limit. Experimental results from gasphase nuclear magnetic resonance of¹²⁹Xe (I=1/2) and¹³¹Xe (I=3/2) are in excellent agreement with theoretical predictions.     

Two-beam linear magneto-optical spectroscopy of atomic transitions between short lived states

Two-beam, linear magneto-optical spectroscopy is a powerful tool for studying short-lived states. We present both measurements and a quantitative theoretical analysis of magneto-rotation observed in the forward scattering of a linearly polarised laser beam passing through an amplifying atomic medium placed in a longitudinal magnetic field. The probed transition connects two short-lived, excited atomic levels, the upper state (here the 7S _1/2 level of cesium) being prepared initially via another transition from ground state, excited by a linearly polarised pump beam. The probe polarisation undergoes three different magneto-optical processes: optical rotation, with separate contributions from the two transitions, and linear dichroism due to Hanle precession of the upper state alignment. Complete resolution of the hyperfine structures and ninety degree switching of the probe polarisation enable us to isolate all of these processes. To lowest order in optical thickness the relative intensities and lineshapes are well interpreted.     

Perturbation theory for degenerate states of atomic and molecular systems

The first few terms of the perturbation expansions for the function and the energy shift of a degenerate state of an arbitrary quantum mechanical system are obtained using the adiabatic formula. It is shown how the expansion for the secular operator may be obtained from the expansion for the function. The results are used to calculate energies of the ground and some excited states and multiplet splittings of some beryllium-like ions.     

Twist dynamics of 9-(N-carbazolyl)-anthracene: effects of Intramolecular Vibrational Redistribution and non-adiabatic transitions in coupled bright and dark states

The twist dynamics of 9-(N-carbazolyl)-anthracene (C9A) in the electronically excited bright and dark states S ₁ and S _X is investigated theoretically. Effects of Intramolecular Vibrational Redistribution (IVR) as well as the diabatic transition S ₁ → S _X are taken into account. Based on a model hamiltonian and IVR-rates, which have been derived earlier from laser induced fluorescence spectra and life times observed by Monte et al. [1], the Liouville-von Neumann equation is integrated with a dissipative Liouvillian in Lindblad form. The resulting time dependent density matrix yields nearly coherent twist dynamics of this rather large molecule, at least on a timescale of several ps. The corresponding pump-probe spectra are predicted following the approach proposed by Manz et al. [2].     

Complete quantum control of the population transfer branching ratio between two degenerate target states

A five-level four-pulse phase-sensitive extended stimulated Raman adiabatic passage scheme is proposed to realize complete control of the population transfer branching ratio between two degenerate target states. The control is achieved via a three-node null eigenstate that can be correlated with an arbitrary superposition of the target states. Our results suggest that complete suppression of the yield of one of two degenerate product states, and therefore absolute selectivity in photochemistry, is achievable and predictable, even without studying the properties of the unwanted product state beforehand.     

Generalized brillouin theorem multiconfiguration method for excited states

The Generalized Brillouin Theorem Multiconfiguration Method (GBT-MC) of Grein and Chang is extended and applied to the calculation of excited states. Orthogonality constraints to lower states as well as second-order interaction effects of states lying close together have been taken into account. In this way quadratic convergence can be guaranteed. Difficulties with coupling coefficients and Lagrangian multipliers of SCF methods can be circumvented. Test calculations have been performed on valence electron excited states of C, H₂O, and CH₂O, and on core excited states of Li.     

Reactivity indicators for degenerate states in the density-functional theoretic chemical reactivity theory

Density-functional-theory-based chemical reactivity indicators are formulated for degenerate and near-degenerate ground states. For degenerate states, the functional derivatives of the energy with respect to the external potential do not exist, and must be replaced by the weaker concept of functional variation. The resultant reactivity indicators depend on the specific perturbation. Because it is sometimes impractical to compute reactivity indicators for a specific perturbation, we consider two special cases: point-charge perturbations and Dirac delta function perturbations. The Dirac delta function perturbations provide upper bounds on the chemical reactivity. Reactivity indicators using the common used "average of degenerate states approximation" for degenerate states provide a lower bound on the chemical reactivity. Unfortunately, this lower bound is often extremely weak. Approximate formulas for the reactivity indicators within the frontier-molecular-orbital approximation and special cases (two or three degenerate spatial orbitals) are presented in the supplementary material. One remarkable feature that arises in the frontier molecular orbital approximation, and presumably also in the exact theory, is that removing electrons sometimes causes the electron density to increase at the location of a negative (attractive) Dirac delta function perturbation. That is, the energetic response to a reduction in the external potential can increase even when the number of electrons decreases.     

Adiabatic perturbation theory for the degenerate case. II. Perturbation of degenerate bound states embedded in the continuum

An adiabatic formula for the contracted Hamiltonian in a reference space containing bound-state eigenfunctions of degenerate energy levels embedded in the continuum is derived. A general factorization theorem for the dynamic operatorS_α(0, – ∞/λ) is proved, and the cancellation of the pole singularities in the perturbation series of the contracted Hamiltonian in adiabatic form is discussed.