Adiabatic Wave Functions Research Articles

On the Topological Phase around Conical Intersections with Tamm-Dancoff Linear-Response Time-Dependent Density Functional Theory.

Regions of nuclear-configuration space away from the Franck-Condon geometry can prove problematic for some electronic structure methods, given the propensity of such regions to possess conical intersections, i.e., (highly connected) points of degeneracy between potential energy surfaces. With the likelihood (perhaps even inevitability) for nonadiabatic dynamics simulations to explore molecular geometries in close proximity to conical intersections, it is vital that the performance of electronic structure methods is routinely examined in this context. In a recent paper [Taylor, J. T. J. Chem. Phys. 2023, 159, 214115.], the ability of linear-response time-dependent density functional theory within the adiabatic approximation (AA LR-TDDFT) to provide a proper description of conical intersections, in terms of their topology and topography, was investigated, with particular attention paid to conical intersections between two excited electronic states. For the same prototypical molecules, protonated formaldimine and pyrazine, we herein consider whether AA LR-TDDFT can correctly reproduce the topological phase accumulated by the adiabatic electronic wave function upon traversing a closed path around an excited-to-excited state conical intersection despite not using the appropriate quadratic-response nonadiabatic coupling vectors. Equally, we probe the ability of the ground-to-excited state intersection ring exhibited by AA LR-TDDFT in protonated formaldimine to give rise to a similar topological phase in spite of its incorrect dimensionality.

Compensation States Approach in the Hybrid Diabatization Scheme: Extension to Multidimensional Data and Properties.

The diabatization of reactive systems for more than just a couple of states is a very demanding problem and generally requires advanced diabatization techniques. Especially for dissociative processes, the drastic changes in the adiabatic wave functions often would require large diabatic state bases, which quickly become impractical. Recently, we addressed this problem by the compensation states approach developed in the context of our hybrid diabatization scheme. This scheme utilizes wave function as well as energy data in combination with a diabatic potential model. In regions where the initial diabatic state basis becomes insufficient for an appropriate representation of the adiabatic states, new model states are generated. The new model states compensate for the state space not spanned by the initial diabatic basis. Such a compensation state is obtained by projecting the initial diabatic state space out of the adiabatic wave function. This yields a very efficient basis representation of the electronic Hamiltonian. The present work presents two new aspects. First, it is shown how other operators like the spin-orbit operator in the framework of the Effective Relativistic Coupling by Asymptotic Representation (ERCAR) can be evaluated in this compact model state space without losing the correct wave function information and accuracy. Second, the extension of the approach to multidimensional potential energy surface models is presented for methyl iodide including the C-I dissociation coordinate and the angular H3C-I bending coordinates.

Phase Regularization and Additional Potential in Quantum Systems at the Potential Barrier to Maintain Adiabatic Condition

This research is theoretical research with a literature study that examines methods to maintain characteristics of electrons when moving in a quantum system at the potential barrier. Attempts to maintain the characteristics of electrons in quantum systems are known as adiabatic. The method used in this research is the fast-forward method. This method was first introduced by Masuda and Nakamura in 2010. The fast-forward method is applied to barrier potential in case of electron energy larger and smaller than the potential barrier. This research focuses on preserving the characteristics of electrons by determining the regularization phase and additional potential of the quantum system at the barrier potential. The wave function solution of the system at the potential barrier is approximated by the Schrödinger equation into three regions for each case. The wave function of each region is regularized to be an adiabatic wave function and an additional term in the form of regularization phase (θ) is obtained. Given the regularized Hamiltonian, the additional potential (Ṽ) is obtained. The obtained regularization phase and additional potential ensure the quantum system at the potential barrier is in the same state from the initial state to the final state

Quantum loop effects on the power spectrum and constraints on primordial black holes

We present a detailed exposition on the prospects of the formation of Primordial Black Holes (PBHs) during Slow Roll (SR) to Ultra Slow Roll (USR) sharp transitions in the framework of single-field inflation. We use an effective field theory (EFT) approach in order to keep the analysis model-independent and applicable to both the canonical and non-canonical cases. We show in detail how renormalizing the power spectrum to one loop order in P(X,ϕ) theories severely limits the prospects for PBH formation in a single-field inflationary framework. We demonstrate that for the allowed range of effective sound speed, 1 < cs < 1.17, the consistency of one-loop corrected power spectrum leaves a small window for black hole masses, M PBH∼𝒪(102–103)gm to have sufficient e-foldings, Δ𝒩Total ∼ 𝒪(54–59) for inflation. We confirm that adding an SR regime after USR before the end of inflation does not significantly alter our conclusions. Our findings for sharp transition strictly rule out the possibility of generating large masses of PBHs from all possible models of single field inflation (canonical and non-canonical).Our results are at least valid for the situation where constraints from the loop effects are computed using either Late-Time (LT) or Adiabatic-Wave function(AF) scheme followed by Power Spectrum (PS) renormalization schemes.

Black-body radiation-induced photodissociation and population redistribution of weakly bound states in H2 +

Molecular hydrogen ions in weakly bound states close to the first dissociation threshold are attractive quantum sensors for measuring the proton-to-electron mass ratio and hyperfine-induced ortho-para mixing. The experimental accuracy of previous spectroscopic studies relying on fast ion beams could be improved by using state-of-the-art ion trap setups. With the electric-dipole moment vanishing in H and preventing fast spontaneous emission, radiative lifetimes of the order of weeks are found. We include the effect of black-body radiation that can lead to photodissociation and rovibronic state redistribution to obtain effective lifetimes for trapped ion experiments. Rate coefficients for bound-bound and bound-continuum processes were calculated using adiabatic nuclear wave functions and nonadiabatic energies, including relativistic and radiative corrections. Effective lifetimes for the weakly bound states were obtained by solving a rate equation model and lifetimes in the range of 4–523 and >215 ms were found at room temperature and liquid nitrogen temperature, respectively. Black-body induced photodissociation was identified as the lifetime-limiting effect, which guarantees the purity of state-selectively generated molecular ion ensembles. The role of hyperfine-induced g/u-mixing, which allows pure rovibrational transitions, was found to be negligible.

Удаан M^{-}+He мөргөлдөөний үед электроны корреляц тооцох асуудал

An adiabatic electronic wave function for helium atoms depending on distance between the heavy particles (the negative meson and the target nuclear) is constructed. Both the radial and angle correlations are included in the wave function and the adiabatic electronic term for the He(1s2)+M- system is calculated. It is shown that the adiabatic electronic term for the elastic (M-+He(1s2) ) and ionization (M-+He(1s)) channels cross twice at the distance R1=0.88a0 and R2=0.215a0 between heavy particles. A possible corrections to the theory for ionization of atoms by slow meson impact, due to the second crossing point (R2), are discussed.

Adiabatic expressions for the wave function of an electron in a finite-range potential and an intense low-frequency laser pulse

The wave function of an electron interacting with a finite-range potential and an intense low-frequency laser pulse is analyzed within the adiabatic approximation. The closed analytic form for the wave function, which includes the rescattering corrections, is obtained with quasiclassical accuracy for an electron in both the initial bound and continuum states. We discuss the parametrizations of amplitudes of fundamental strong-field processes in terms of laser and binding-potential parameters. Based on the analytic results for the adiabatic wave functions, we develop the perturbation theory in an additional weak field. The modification of high-order harmonic generation amplitude caused by a weak extreme ultraviolet pulse is discussed in the first order of the perturbation theory.

Towards multistate multimode landscapes in singlet fission of pentacene: the dual role of charge-transfer states.

Singlet fission duplicates triplet excitons for improving light harvesting efficiency. The presence of the interaction between electronic and nuclear degrees of freedom complicates the interpretation of correlated triplet pairs. We report a quantum chemistry study on the significance and subtleties of multistate and multimode pathways in forming triplet pair states of the pentacene dimer through a six-state vibronic-coupling Hamiltonian derived from many-electron adiabatic wavefunctions of an ab initio density matrix renormalization group. The resulting spin values of the singlet manifolds on each pentacene center are computed, and the varying spin nature can be distinguished clearly with respect to dimer stacking and vibronic progression. Our monomer spin assignments reveal the coexistence of both lower-lying weak and higher-lying strong charge transfer states which interact vibronically with the triplet pair state, providing important implications for its generation and separation occurring in vibronic regions. This work conveys the importance of the many-electron process requiring close low-lying singlet manifolds to determine the subtle fission details, and represents an important step for understanding vibronically resolved spin states and conversions underlying efficient singlet fission.

WKB and “cubic-WKB” methods as an adiabatic approximation

This paper shows that WKB wave function can be expressed in the form of an adiabatic expansion. To build a bridge between two widely invoked approximation schemes seems pedagogically instructive. Further, “cubic-WKB” method that has been devised in order to overcome the divergence problem of WKB can be also presented in the form of an adiabatic approximation: The adiabatic expansion of a wave function contains a certain parameter. When this parameter is adjusted so as to make the next order correction vanish approximately, the adiabatic wave function becomes equivalent to that of the “cubic-WKB”.

Solving the Trivial Crossing Problem While Preserving the Nodal Symmetry of the Wave Function.

In an adiabatic mixed quantum-classical simulation, the avoided crossing of weakly coupled eigenstates can lead to unphysical discontinuities in wave function dynamics, otherwise known as the trivial crossing problem. A standard solution to the trivial crossing problem eliminates spatial discontinuities in wave function dynamics by imposing changes to the eigenstate of the wave function. In this paper, we show that this solution has the side effect of introducing transient discontinuities in the nodal symmetry of the wave function. We present an alternative solution to the trivial crossing problem that preserves both the spatial and nodal structure of the adiabatic wave function. By considering a model of exciton dynamics on conjugated polymer systems, we show that failure to preserve wave function symmetry yields exciton dynamics that depends unphysically on polymer system size. We demonstrate that our symmetry-preserving solution to the trivial crossing problem yields more realistic dynamics and can thus improve the accuracy of simulations of larger systems that are prone to the trivial crossing problem.

Optical Absorption by Charge-Transfer Molecules.

An analytical formalism is developed for the calculation of absorption of radiation by charge-transfer molecules with electronic density delocalized between the donor and acceptor parts. The theory consistently incorporates both the vibronic coupling to quantum intramolecular vibrations and electrostatic interactions with a classical polarizable medium. The formulation operates in terms of basis-invariant parameters and can be used for calculations based on both the localized diabatic states and delocalized adiabatic wave function produced by standard quantum-chemistry algorithms. The basis-invariant reorganization energy is particularly important, since it determines observable spectroscopic parameters, in contrast to the adiabatic reorganization energy based on the molecular charge distributions in the adiabatic vacuum states. Analytical formulas are derived for the charge-transfer absorption band and first two spectral moments connecting spectroscopy to properties of charge-transfer molecules. Electronic delocalization requires a new definition of the Huang-Rhys factor traditionally used to construct vibronic band shapes. Theory's performance is illustrated by application to experimental data for self-exchange charge-transfer optical absorption and for metal-to-ligand emission transitions.

Up to a Sign. The Insidious Effects of Energetically Inaccessible Conical Intersections on Unimolecular Reactions.

It is now well established that conical intersections play an essential role in nonadiabatic radiationless decay where their double-cone topography causes them to act as efficient funnels channeling wave packets from the upper to the lower adiabatic state. Until recently, little attention was paid to the effect of conical intersections on dynamics on the lower state, particularly when the total energy involved is significantly below that of the conical intersection seam. This energetic deficiency is routinely used as a sufficient condition to exclude consideration of excited states in ground state dynamics. In this account, we show that, this energy criterion notwithstanding, energy inaccessible conical intersections can and do exert significant influence on lower state dynamics. The origin of this influence is the geometric phase, a signature property of conical intersections, which is the fact that the real-valued electronic wave function changes sign when transported along a loop containing a conical intersection, making the wave function double-valued. This geometric phase is permitted by an often neglected property of the real-valued adiabatic electronic wave function; namely, it is determined only up to an overall sign. Noting that in order to change sign a normalized, continuous function must go through zero, for loops of ever decreasing radii, demonstrating the need for an electronic degeneracy (intersection) to accompany the geometric phase. Since the total wave function must be single-valued a compensating geometry dependent phase needs to be included in the total electronic-nuclear wave function. This Account focuses on how this consequence of the geometric phase can modify nuclear dynamics energetically restricted to the lower state, including tunneling dynamics, in directly measurable ways, including significantly altering tunneling lifetimes, thus confounding the relation between measured lifetimes and barrier heights and widths, and/or completely changing product rotational distributions. Some progress has been made in understanding the origin of this effect. It has emerged that for a system where the lower adiabatic potential energy surface exhibits a topography comprised of two saddle points separated by a high energy conical intersection, the effect of the geometric phase can be quite significant. In this case topologically distinct paths through the two adiabatic saddle points may lead to interference. This was pointed out by Mead and Truhlar almost 50 years ago and denoted the Molecular Aharonov-Bohm effect. Still, the difficulty in anticipating a significant geometric phase effect in tunneling dynamics due to energetically inaccessible conical intersections leads to the attribute insidious that appears in the title of this Account. Since any theory is only as relevant as the prevalence of the systems it describes, we include in this Account examples of real systems where these effects can be observed. The accuracy of the reviewed calculations is high since we use fully quantum mechanical dynamics and construct the geometric phase using an accurate diabatic state fit of high quality ab initio data, energies, energy gradients, and interstate couplings. It remains for future work to establish the prevalence of this phenomenon and its deleterious effects on the conventional wisdom discussed in this work.

A discontinuous basis enables numerically exact solution of the Schrödinger equation around conical intersections in the adiabatic representation.

Solving the vibrational Schrödinger equation in the neighborhood of conical intersections in the adiabatic representation is a challenge. At the intersection point, first- and second-derivative nonadiabatic coupling matrix elements become singular, with the singularity in the second-derivative coupling (diagonal Born-Oppenheimer correction) being non-integrable. These singularities result from discontinuities in the vibronic functions associated with the individual adiabatic states, and our group has recently argued that these divergent matrix elements cancel when discontinuous adiabatic vibronic functions sum to a continuous total nonadiabatic wave function. Here we describe the realization of this concept: a novel scheme for the numerically exact solution of the Schrödinger equation in the adiabatic representation. Our approach is based on a basis containing functions that are discontinuous at the intersection point. We demonstrate that the individual adiabatic nuclear wave functions are themselves discontinuous at the intersection point. This proves that discontinuous basis functions are essential to any tractable method that solves the Schrödinger equation around conical intersections in the adiabatic representation with high numerical precision. We establish that our method provides numerically exact results by comparison to reference calculations performed in the diabatic representation. In addition, we quantify the energetic error associated with constraining the density to be zero at the intersection point, a natural approximation. Prospects for extending the present treatment of a two-dimensional model to systems of higher dimensionality are discussed.

Interstate vibronic coupling constants between electronic excited states for complex molecules.

In the construction of diabatic vibronic Hamiltonians for quantum dynamics in the excited-state manifold of molecules, the coupling constants are often extracted solely from information on the excited-state energies. Here, a new protocol is applied to get access to the interstate vibronic coupling constants at the time-dependent density functional theory level through the overlap integrals between excited-state adiabatic auxiliary wavefunctions. We discuss the advantages of such method and its potential for future applications to address complex systems, in particular, those where multiple electronic states are energetically closely lying and interact. We apply the protocol to the study of prototype rhenium carbonyl complexes [Re(CO)3(N,N)(L)]n+ for which non-adiabatic quantum dynamics within the linear vibronic coupling model and including spin-orbit coupling have been reported recently.

Diabatic states, nonadiabatic coupling, and the counterpoise procedure for weakly interacting open-shell molecules

We study nonadiabatic coupling in systems of weakly interacting open-shell molecules which have nearly degenerate electronic states and hence significant nuclear derivative couplings. By comparison to numerically calculated nuclear derivatives of adiabatic electronic wave functions, we show that nonadiabatic couplings are represented accurately by diabatization using a recent multiple-property-based algorithm [T. Karman et al., J. Chem. Phys. 144, 121101 (2016)]. Accurate treatment of weakly interacting molecules furthermore requires counterpoise corrections for the basis-set superposition error. However, the generalization of the counterpoise procedure to open-shell systems is ambiguous. Various generalized counterpoise schemes that have been proposed previously are shown to be related through different choices for diabatization of the monomer wave functions. We compare these generalized counterpoise schemes and show that only two approaches accurately describe long-range interactions. In addition, we propose an approximate diabatization algorithm based on the asymptotic long-range interaction. This approach is appealingly simple to implement as it yields analytical expressions for the transformation to the diabatic representation. Finally, we investigate the effects of diabatizing intermolecular potentials on the nuclear dynamics by performing quantum scattering calculations for NO(X2Π)–H2. We show that cross sections for pure rotational transitions are insensitive to diabatization. For spin-orbit inelastic transitions, asymptotic diabatization and multiple-property-based diabatization are in qualitative agreement, but the quantitative differences may be observable experimentally.

Geometric Phase Effects in Nonadiabatic Dynamics near Conical Intersections

Dynamical consideration that goes beyond the common Born-Oppenheimer approximation (BOA) becomes necessary when energy differences between electronic potential energy surfaces become small or vanish. One of the typical scenarios of the BOA breakdown in molecules beyond diatomics is a conical intersection (CI) of electronic potential energy surfaces. CIs provide an efficient mechanism for radiationless electronic transitions: acting as "funnels" for the nuclear wave function, they enable rapid conversion of the excessive electronic energy into the nuclear motion. In addition, CIs introduce nontrivial geometric phases (GPs) for both electronic and nuclear wave functions. These phases manifest themselves in change of the wave function signs if one considers an evolution of the system around the CI. This sign change is independent of the shape of the encircling contour and thus has a topological character. How these extra phases affect nonadiabatic dynamics is the main question that is addressed in this Account. We start by considering the simplest model providing the CI topology: two-dimensional two-state linear vibronic coupling model. Selecting this model instead of a real molecule has the advantage that various dynamical regimes can be easily modeled in the model by varying parameters, whereas any fixed molecule provides the system specific behavior that may not be very illustrative. After demonstrating when GP effects are important and how they modify the dynamics for two sets of initial conditions (starting from the ground and excited electronic states), we give examples of molecular systems where the described GP effects are crucial for adequate description of nonadiabatic dynamics. Interestingly, although the GP has a topological character, the extent to which accounting for GPs affect nuclear dynamics profoundly depends on topography of potential energy surfaces. Understanding an extent of changes introduced by the GP in chemical dynamics poses a problem of capturing GP effects by approximate methods of simulating nonadiabatic dynamics that can go beyond simple models. We assess the performance of both fully quantum (wave packet dynamics) and quantum-classical (surface-hopping, Ehrenfest, and quantum-classical Liouville equation) approaches in various cases where GP effects are important. It has been identified that the key to success in approximate methods is a method organization that prevents the quantum nuclear kinetic energy operator to act directly on adiabatic electronic wave functions.

Constructive and Destructive Interference in Nonadiabatic Tunneling via Conical Intersections.

As a manifestation of the molecular Aharonov-Bohm effect, tunneling-facilitated dissociation under a conical intersection (CI) requires the inclusion of the geometric phase (GP) to ensure a single-valued adiabatic wave function encircling the CI. In this work, we demonstrate using a simple two-dimensional model that the GP induces destructive interference for vibrational states with even quanta in the coupling mode, but it leads to constructive interference for those with odd quanta. The interference patterns are manifested in tunneling wave functions and clearly affect the tunneling lifetime. It is further shown that the inclusion of the diagonal Born-Oppenheimer correction is necessary for agreement with exact results.

Nonadiabatic tunneling via conical intersections and the role of the geometric phase

As a ubiquitous quantum effect, tunneling has attracted attention ever since the dawn of quantum mechanics. However, recent evidence suggests that nonadiabatic atomic tunneling near a conical intersection (CI) behaves differently from its adiabatic counterpart, producing lifetime differences of up to two orders of magnitude. Using two-dimensional models, we demonstrate here that the failure of the adiabatic model in describing tunneling near a CI can be attributed largely to the neglect of the geometric phase, which is associated with the adiabatic electronic wave function transported around a CI. The geometric phase--induced destructive interference among wave functions following different paths around the CI, manifested as a node in the adiabatic wave function, retards tunneling.

On the incorporation of the geometric phase in general single potential energy surface dynamics: A removable approximation to ab initio data.

For two electronic states coupled by conical intersections, the line integral of the derivative coupling can be used to construct a complex-valued multiplicative phase factor that makes the real-valued adiabatic electronic wave function single-valued, provided that the curl of the derivative coupling is zero. Unfortunately for ab initio determined wave functions, the curl is never rigorously zero. However, when the wave functions are determined from a coupled two diabatic state Hamiltonian Hd (fit to ab initio data), the resulting derivative couplings are by construction curl free, except at points of conical intersection. In this work we focus on a recently introduced diabatization scheme that produces the Hd by fitting ab initio determined energies, energy gradients, and derivative couplings to the corresponding Hd determined quantities in a least squares sense, producing a removable approximation to the ab initio determined derivative coupling. This approach and related numerical issues associated with the nonremovable ab initio derivative couplings are illustrated using a full 33-dimensional representation of phenol photodissociation. The use of this approach to provide a general framework for treating the molecular Aharonov Bohm effect is demonstrated.

A new approach for the development of diabatic potential energy surfaces: Hybrid block-diagonalization and diabatization by ansatz.

A new diabatization method is presented, which is suitable for the development of accurate high-dimensional coupled potential energy surfaces for use in quantum dynamics studies. The method is based on the simultaneous use of adiabatic wave function and energy data, respectively, and combines block-diagonalization and diabatization by ansatz approaches. It thus is called hybrid diabatization. The adiabatic wave functions of suitable ab initio calculations are projected onto a diabatic state space and the resulting vectors are orthonormalized like in standard block-diagonalization. A parametrized diabatic model Hamiltonian is set up as an ansatz for which the block-diagonalization data can be utilized to find the optimal model. Finally, the parameters are optimized with respect to the ab initio reference data such that the deviations between adiabatic energies and eigenvalues of the model as well as projected state vectors and eigenvectors of the model are minimized. This approach is particularly advantageous for problems with a complicated electronic structure where the diabatic state space must be of higher dimension than the number of calculated adiabatic states. This is an efficient way to handle problems with intruder states, which are very common for reactive systems. The use of wave function information also increases the information content for each data point without additional cost, which is beneficial in handling the undersampling problem for high-dimensional systems. The new method and its performance are demonstrated by application to three prototypical systems, ozone (O3), methyl iodide (CH3I), and propargyl (H2CCCH).