Curl equations as substratum for the derivation of the electronic nonadiabatic coupling terms

The idea to derive the nonadiabatic coupling terms by solving the Curl equations (Avery, J.; Baer, M.; Billing, G. D. Mol Phys 2002, 100, 1011) is extended to a three‐state system where the first and second states form one conical intersection, i.e., τ₁₂ and the second and the third states form another conical intersection, i.e., τ₂₃. Whereas the two‐state Curl equations form a set of linear differential equations, the extension to a three‐state system not only increases the number of equations but also leads to nonlinear terms. In the present study, we developed a perturbative scheme, which guarantees convergence if the overlap between the two interacting conical intersections is not too strong. Among other things, we also revealed that the nonadiabatic coupling term between the first and third states, i.e., τ₁₃ (such interactions do not originate from conical intersection) is formed due to the interaction between τ₁₂ and τ₂₃. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Space-time contours to treat the interaction between an intense electric field and a molecular system

A few years ago, we developed an approach to treat molecular systems exposed to an external, intense, time-dependent field (J. Phys. Chem. A 2003, 107, 4724; J. Chem. Phys. 2003, 119, 6998). Within this study, we encountered two novel concepts: the dressed (namely, field affected) time-dependent nonadiabatic coupling term and the space-time contours. In the present article, we analyze the newly introduced nonadiabatic coupling term and discuss its importance for dynamical studies. We also refer to the just mentioned space-time contour and present the more efficient contour for realistic situations. The scope of the above-mentioned articles is extended with the aim of defining quasi-adiabatic states for such situations. Strictly speaking, molecular systems exposed to intense, fast oscillating fields are not expected to form adiabatic states. Still we consider such a situation and end up with three possibilities for quasi-adiabatical time-dependent states eventually to be used within the Born-Oppenheimer approximation.     

A field theoretical approach to calculate electronic Born-Oppenheimer coupling terms

In this paper we suggest to consider the spatial distribution of the Born-Oppenheimer nonadiabatic coupling terms as fields which are created by sources, located at degeneracy points, and which can be derived using the ordinary mathematical tools of field theory. It is shown that the curl-divergence equations as formed within a given Hilbert space [M. Baer, Chem. Phys. Lett. 35, 112 (1975)] can be converted into a set of inhomogeneous coupled Poisson equations which are solved for a given set of boundary conditions. The method is applied to the three-state Hilbert subspace of the H(3) system. The numerical results are compared with ab initio calculations for which a very encouraging fit is found.     

Numerical coupled Liouville approach: Application to second hyperpolarizability of molecular aggregate

In our previous study [Int. J. Quant. Chem., to appear], we have developed a novel numerical calculation scheme for a dynamics of quantum network for linear molecular aggregates under intense time‐dependent electric fields. In this approach, each molecule is assumed to be an electric dipole arranged linearly with an angle from the longitudinal axis, and the molecular interactions are taken into account by adding the radiations from these dipoles to the external electric fields. The effects of the radiations from all the dipoles involve the intermolecular distance, the speed of light, retarded polarization, and its first‐ and second‐order time derivatives at the position of each dipole. The quantum dynamics is performed by solving coupled Liouville equations composed of the Liouville equation for each dipole. In the present study, we develop a calculation approach of nonperturbative second hyperpolarizability γ in our novel approach and examine the γ of dimer models composed of two‐state molecules under the one‐photon near resonant intense laser fields. Similar phase transition‐like behavior in the field‐intensity dependence of the γ is observed. We also investigate the second hyperpolarizability spectra in the three‐photon resonant region for dimers composed of three‐state molecules, which mimic the electronic states of allyl cation. Contrary to the one‐photon resonant case, phase transition‐like behavior is not observed in the intensity dependence of γ in the three‐photon resonant region. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 295–306, 1999     

Derivation of the electronic nonadiabatic coupling field in molecular systems: an algebraic-vectorial approach

In this Communication it is suggested that various elements of the nonadiabatic coupling matrix, tau(jk)(s) are created by the singular nonadiabatic coupling terms of the system. Moreover, given the spatial distribution of these coupling terms in the close vicinity of their singularity points yields, according to this approach, the integrated intensity of the field at every point in the region of interest. To support these statements we consider the conical intersections of the three lower states of the H+H(2) system: From an ab initio treatment we obtain the nonadiabatic coupling terms around each conical intersection separately (at its close vicinity) and having those, create the field at every desired point employing vector-algebra. This approach is also used to calculate the intensity of the Curl of those matrix elements that lack their own sources [tau(13)(s) in the present case]. The final results are compared with relevant ab initio calculations.     

Density‐matrix theory of quantum dynamics under a strong external field switched on nonadiabatically

Quantum time‐evolution equations for the density matrix are formulated in the unrestricted Hartree–Fock approximation, with an emphasis on the nonperturbative effect due to a sudden or gradual onset of a strong external field. Numerical simulations are performed for ideal Fermi gas around a square‐well potential which is switched on dynamically. When the switching is fast enough, an oscillatory motion of the particle is induced by a nonadiabatic transition at the Landau–Zener crossing point, which is most clearly seen in a small‐size system. When the switching is sufficiently slow, the simulation corroborates the adiabatic theorem. It is shown that the Anderson's infrared catastrophe in a metal is strongly enhanced by the nonperturbative effect. The Keldysh formula of atomic multiphoton ionization can also be derived from the nonperturbative term in the density‐matrix equation, indicating a wide applicability of the present theory. © 2014 Wiley Periodicals, Inc.     

Nonadiabatic electron wavepacket dynamics of molecules in an intense optical field: an ab initio electronic state study

A theory of quantum electron wavepacket dynamics that nonadiabatically couples with classical nuclear motions in intense optical fields is studied. The formalism is intended to track the laser-driven electron wavepackets in terms of the linear combination of configuration-state functions generated with ab initio molecular orbitals. Beginning with the total quantum Hamiltonian for electrons and nuclei in the vector potential of classical electromagnetic field, we reduce the Hamiltonian into a mixed quantum-classical representation by replacing the quantum nuclear momentum operators with the classical counterparts. This framework gives equations of motion for electron wavepackets in an intense laser field through the time dependent variational principle. On the other hand, a generalization of the Newtonian equations provides a matrix form of forces acting on the nuclei for nonadiabatic dynamics. A mean-field approximation to the force matrix reduces this higher order formalism to the semiclassical Ehrenfest theory in intense optical fields. To bring these theories into a practical quantum chemical package for general molecules, we have implemented the relevant ab initio algorithms in it. Some numerical results in the level of the semiclassical Ehrenfest-type theory with explicit use of the nuclear kinematic (derivative) coupling and the velocity form for the optical interaction are presented.     

利用基于直接动力学的轨线面跳跃方法处理非绝热过程

势能面交叉引起的非绝热过程广泛存在于光化学和光物理中。对这一过程进行描述是理论化学的重要挑战之一。非绝热过程涉及原子核与电子之间的耦合运动,因此量子化学的基本假设之一"玻恩-奥本海默"近似被打破,所以对其进行描述需要发展新的动力学理论方法。在这些方法中,Tully发展的最少轨线面跳跃方法凭借易于程序化、便于计算等优点已经发展成为处理非绝热问题的主要动力学方法之一。其中原子核以经典的方式在单一势能面上进行演化,电子以量子的方式沿着同一轨线进行演化。在整个演化过程中,非绝热跃迁通过轨线在不同势能面间的跃迁来描述,其中跳跃发生的几率与电子的演化有关。如果将该方法与从头算直接动力学相结合,可以在全原子水平上研究实际分子体系的非绝热动力学,给出其激发态寿命、非绝热动力学中分子的主要运动方式、反应通道以及分支比等重要信息。本文旨在讨论最少面跳跃直接动力学方法研究非绝热问题的一些进展,包括动力学基本理论,特别关注将最少面跳跃方法和直接动力学结合的数值实现细节,同时讨论该方法在研究实际体系当中的一些应用,并对轨线面跳跃方法下一步发展的一些方向进行合理的展望。     

Quantum fluctuation of electronic wave-packet dynamics coupled with classical nuclear motions

An ab initio electronic wave-packet dynamics coupled with the simultaneous classical dynamics of nuclear motions in a molecule is studied. We first survey the dynamical equations of motion for the individual components. Reflecting the nonadiabatic dynamics that electrons can respond to nuclear motions only with a finite speed, the equations of motion for nuclei include a force arising from the kinematic (nuclear momentum) coupling from electron cloud. To materialize these quantum effects in the actual ab initio calculations, we study practical implementation of relevant electronic matrix elements that are related to the derivatives with respect to the nuclear coordinates. Applications of the present scheme are performed in terms of the configuration state functions (CSF) using the canonical molecular orbitals as basis functions without transformation to particular diabatic basis. In the CSF representation, the nonadiabatic interaction due to the kinematic coupling is anticipated to be rather small, and instead it should be well taken into account through the off-diagonal elements of the electronic Hamiltonian matrix. Therefore it is expected that the nonadiabatic dynamics based on this CSF basis neglecting the kinematic coupling may work. To verify this anticipation and to quantify the actual effects of the kinematic coupling, we compare the dynamics with and without the kinematic-coupling terms using the same CSF set. Applications up to the fifth electronically excited states in a nonadiabatic collision between H(2) and B(+) shows that the overall behaviors of these two calculations are surprisingly similar to each other in an average sense except for a fast fluctuation reflecting the electronic time scale. However, at the same time, qualitative differences in the collision events are sometimes observed. Therefore it turns out after all that the kinematic-coupling terms cannot be neglected in the CSF-basis representation. The present applications also demonstrate that the nonadiabatic electronic wave-packet dynamics within ab initio quantum chemical calculation is feasible.     

Curl condition for a four-state Born-Oppenheimer system employing the Mathieu equation

When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions--the necessary requirements to ensure the adiabatic-diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born-Oppenheimer (EBO) equations [J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.     

Numerical examination of the extended phase‐space volume‐preserving integrator by the Nosé‐Hoover molecular dynamics equations

This article illustrates practical applications to molecular dynamics simulations of the recently developed numerical integrators [Phys Rev E 2006, 73, 026703] for ordinary differential equations. This method consists of extending any set of ordinary differential equations in order to define a time invariant function, and then use the techniques of divergence‐free solvable decomposition and symmetric composition to obtain volume‐preserving integrators in the extended phase space. Here, we have developed the technique by constructing multiple extended‐variable formalism in order to enhance the handling in actual simulation, and by constituting higher order integrators to obtain further accuracies. Using these integrators, we perform constant temperature molecular dynamics simulations of liquid water, liquid argon and peptide in liquid water droplet. The temperature control is obtained through an extended version of the Nosé‐Hoover equations. Analyzing the effects of the simulation conditions including time step length, initial values, boundary conditions, and equation parameters, we investigate local accuracy, global accuracy, computational cost, and sensitivity along with the sampling validity. According to the results of these simulations, we show that the volume‐preserving integrators developed by the current method are more effective than traditional integrators that lack the volume‐preserving property. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Ab initio surface hopping excited-state molecular dynamics approach on the basis of spin–orbit coupled states: An application to the A-band photodissociation of CH3I

Ab initio molecular dynamics approach has been extended to multi-state dynamics on the basis of the spin–orbit coupled electronic states that are obtained through diagonalization of the spin–orbit coupling matrix with the multi-state second-order multireference perturbation theory energies in diagonal elements and the spin–orbit coupling terms at the state-averaged complete active space self-consistent field level in off-diagonal elements. Nonadiabatic transitions over the spin–orbit coupled states were taken into account explicitly by a surface hopping scheme with utilizing the nonadiabatic coupling terms calculated by numerical differentiation of the spin–orbit coupled wavefunctions and analytical nonadiabatic coupling terms. The present method was applied to the A-band photodissociation of methyl iodide, CH₃I + hv → CH₃ + I (²P_3/2)/I* (²P_1/2), for which a pioneering theoretical work was reported by Amatatsu, Yabushita, and Morokuma. The present results reproduced well the experimental branching ratio and energy distributions in the dissociative products. © 2018 Wiley Periodicals, Inc.     

Nuclear magnetic resonance J coupling constant polarizabilities of hydrogen peroxide: A basis set and correlation study

In this article, we present the so far most extended investigation of the calculation of the coupling constant polarizability of a molecule. The components of the coupling constant polarizability are derivatives of the nuclear magnetic resonance (NMR) indirect nuclear spin–spin coupling constant with respect to an external electric field and play an important role for both chiral discrimination and solvation effects on NMR coupling constants. In this study, we illustrate the effects of one‐electron basis sets and electron correlation both at the level of density functional theory as well as second‐order polarization propagator approximation for the small molecule hydrogen peroxide, which allowed us to perform calculations with the largest available basis sets optimized for the calculation of NMR coupling constants. We find a systematic but rather slow convergence with the one‐electron basis set and that augmentation functions are required. We observe also large and nonsystematic correlation effects with significant differences between the density functional and wave function theory methods. © 2012 Wiley Periodicals, Inc.     

Effects of disordered interchain interactions on polaron dynamics in semiconducting polymers

Polaron dynamics in a system of two randomly coupled polymer chains is simulated using a nonadiabatic evolution method. The simulations are performed within the framework of the Su-Schrieffer-Heeger model modified to include disordered interchain interactions and an external electric field. By analysing the polaron velocity statistically, we find that the polaron motion is determined by the competition between the electric field and the disordered interchain interactions. Polaron dynamics are classified into two types, weak-coupling dynamics and strong-coupling dynamics. It is found that the strength of interchain interactions is the dominant factor controlling charge propagation in weak-coupling dynamics, whereas the effects of disorder are dominant in strong-coupling dynamics. The charge carriers tend to have higher mobility for stronger interchain coupling, and interchain coupling disorder can be favorable for charge transport depending on the coupling strength and the electric field.     

Two types of Kramers rate equations for reactions in condensed media

Two cases of bilinear coupling of a particle to a medium are compared. They differ in that one of the mediums does not modify the particle potential, whereas the other does. Two corresponding kramers-type rate equations for adiabatic reactions are derived directly from a dynamic rate theory and interpreted from the view of the stochastic theory. Both equations become identical in the weak coupling limit. Their relations to transition-state theory are also discussed. Corresponding results for nonadiabatic reactions are considered. © 1994 John Wiley & Sons, Inc.     

Simultaneous stochastic effects of a thermal reservoir and an electromagnetic field on the optical susceptibility of a two-level system

In this paper we derive analytical expressions for the optical susceptibility of a two-level system immersed in a thermal bath and interacting with an external electromagnetic field, where both of them are considered as noise sources. The dynamics of the system is described by a set of optical stochastic Bloch equations. The noise sources are modeled as Ornstein-Uhlenbeck processes. The optical stochastic Bloch equations are perturbatively solved up to second order in the external field. We found that each noise affects the dynamics in a different manner. Thus, at first order, the bath modifies the transverse relaxation time, whereas the effect of a random field can only be appreciated if the expansion is calculated up to second order, where correlations begin to be important.     

Nonadiabatic scattering of NO off Au3 clusters: A simple and robust diabatic state manifold generation method for multiconfigurational wavefunctions

The presence of nonadiabatic effects during the interaction of small molecules with metals has been observed experimentally for the last decades. Specially remarkable are the effects found for NO/Au, where experiments have suggested the presence of very strong vibronic coupling during the molecular scattering. However, the accurate inclusion of the nonadiabatic effects in periodic boundary conditions (PBC) theoretical methods remain an unapproachable challenge. Here, aiming to give some theoretical insight to the strong vibronic coupling, we have adopted a pragmatic point of view, taking use of an auxiliary simplified system, NO/Au₃. We show the importance of nonadiabatic coupling, during the scattering of NO from a Au₃ cluster, using a diabatic representation of 12 electronic states of the system, including a few charge-transfer states. Our diabatic representation is obtained by rotating the orbital and configuration interaction (CI) vectors of a restricted active space (RAS) wavefunction. We present a strategy for extracting the best effective manifold of states relevant to the system, below some prescribed energy, directly from the RAS CI vectors. This scheme is able to disentangle a large dense manifold of adiabatic states with strong coupling and crossings. This approach is also shown to work for multireference configuration interaction (MRCI). By performing quantum propagations, we observed an increase in vibrational redistribution with increasing initial vibrational or translational energies. We suggest that these nonadiabatic effects should also be present at smaller energies in larger clusters. © 2018 Wiley Periodicals, Inc.     

Quantum equations for vibrational dynamics on metal surfaces

A first principles treatment of the vibrational dynamics of molecular chemisorbates on metal surfaces is presented. It is shown that the mean field quantum evolution of the vibrational position operator is determined by a quantum Langevin equation with an electronic friction. In the mean field limit, the quantum noise and friction are related by the quantum fluctuation-dissipation theorem. The classical limit of this model is shown to agree with previously proposed models. A criterion is presented to describe the validity of the weak-coupling approximation and equations of motion for the dynamics in the presence of strong nonadiabatic coupling to electron-hole pairs are presented.     

Conical intersections in solution: formulation, algorithm, and implementation with combined quantum mechanics/molecular mechanics method

The significance of conical intersections in photophysics, photochemistry, and photodissociation of polyatomic molecules in gas phase has been demonstrated by numerous experimental and theoretical studies. Optimization of conical intersections of small- and medium-size molecules in gas phase has currently become a routine optimization process, as it has been implemented in many electronic structure packages. However, optimization of conical intersections of small- and medium-size molecules in solution or macromolecules remains inefficient, even poorly defined, due to large number of degrees of freedom and costly evaluations of gradient difference and nonadiabatic coupling vectors. In this work, based on the sequential quantum mechanics and molecular mechanics (QM/MM) and QM/MM-minimum free energy path methods, we have designed two conical intersection optimization methods for small- and medium-size molecules in solution or macromolecules. The first one is sequential QM conical intersection optimization and MM minimization for potential energy surfaces; the second one is sequential QM conical intersection optimization and MM sampling for potential of mean force surfaces, i.e., free energy surfaces. In such methods, the region where electronic structures change remarkably is placed into the QM subsystem, while the rest of the system is placed into the MM subsystem; thus, dimensionalities of gradient difference and nonadiabatic coupling vectors are decreased due to the relatively small QM subsystem. Furthermore, in comparison with the concurrent optimization scheme, sequential QM conical intersection optimization and MM minimization or sampling reduce the number of evaluations of gradient difference and nonadiabatic coupling vectors because these vectors need to be calculated only when the QM subsystem moves, independent of the MM minimization or sampling. Taken together, costly evaluations of gradient difference and nonadiabatic coupling vectors in solution or macromolecules can be reduced significantly. Test optimizations of conical intersections of cyclopropanone and acetaldehyde in aqueous solution have been carried out successfully.     

NO2^-/NO2体系电子转移的外电场效应和温度效应

用半经典模型研究NO2^-/NO2体系自交换电子转移的反应机理和速率常数, 探讨了外电场对反应过程势能面, 反应能垒及电子转移速率常数的影响, 确定了能垒崩溃的阈值; 讨论了低能垒条件下电子转移反应的速率常数随温度变化的特征。研究表明, 一定方向的外电场能显著降低电子转移反应的活化能垒并提高反应速率常数k, 而对非绝热电子转移反应, 当温度T和活能能垒Ec满足T=2Ec/R时, k取得极大值。