Determining quasidiabatic coupled electronic state Hamiltonians using derivative couplings: A normal equations based method

The anion photoelectron spectrum and diabatization of tetrazolyl

The potential energy surface of tetrazolyl [cyclic (N4CH)] has a conical intersection seam between the two lowest-energy electronic states near the ground state minimum geometry. This work treats that molecule. The potential energy surfaces used in this study are based on a least-squares fitting procedure that includes ab initio energies, energy gradients, and derivative couplings described using polynomials up to fourth-order and ab initio data obtained from multireference configuration interaction wave functions. A five-electronic-state description was generated with a root mean square absolute energy error of 9.6 cm-1, compared to 326.8 cm-1 when only second-order terms were used. The time-independent multimode vibronic coupling in the KDC approximation was used to simulate and analyze the anion ultraviolet photoelectron spectrum of tetrazolide.

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.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Diabatic States of Molecules

Quantitative simulations of electronically nonadiabatic molecular processes require both accurate dynamics algorithms and accurate electronic structure information. Direct semiclassical nonadiabatic dynamics is expensive due to the high cost of electronic structure calculations, and hence it is limited to small systems, limited ensemble averaging, ultrafast processes, and/or electronic structure methods that are only semiquantitatively accurate. The cost of dynamics calculations can be made manageable if analytic fits are made to the electronic structure data, and such fits are most conveniently carried out in a diabatic representation because the surfaces are smooth and the couplings between states are smooth scalar functions. Diabatic representations, unlike the adiabatic ones produced by most electronic structure methods, are not unique, and finding suitable diabatic representations often involves time-consuming nonsystematic diabatization steps. The biggest drawback of using diabatic bases is that it can require large amounts of effort to perform a globally consistent diabatization, and one of our goals has been to develop methods to do this efficiently and automatically. In this Feature Article, we introduce the mathematical framework of diabatic representations, and we discuss diabatization methods, including adiabatic-to-diabatic transformations and recent progress toward the goal of automatization.

Unified Description of the Jahn-Teller Effect in Molecules with Only Cs Symmetry: Cyclohexoxy in Its Full 48-Dimensional Internal Coordinates

The two lowest potential energy surfaces of cyclohexoxy which are coupled by conical intersections and the spin-orbit interaction are determined in the full 48-dimensional internal coordinate space using a feedforward neural network to fit a diabatic potential energy matrix. The electronic structure data are obtained at the multireference configuration interaction with single- and double-excitation level. Underlying parallels between these coupled surfaces and those of the alkoxy radicals methoxy and isopropoxy are established. Earlier work by Dillon and Yarkony is extended. While the parallels would have been challenging to appreciate using the concept of the Jahn-Teller active modes, they are readily seen in terms of two internal modes centered at the conical intersection: g the energy difference gradient vector and h the interstate coupling gradient vector. In other words, g and h vectors provide a unified description of the Jahn-Teller effect in molecules exhibiting C3v and quasi-C3v symmetries. A spectral simulation in the full 48-vibrational-internal coordinate space is reported. This spectrum is obtained using recently developed algorithms designed to increase the size of the systems that can be treated with a time-independent vibronic coupling approach.

Constructing diabatic representations using adiabatic and approximate diabatic data--Coping with diabolical singularities

We have recently introduced a diabatization scheme, which simultaneously fits and diabatizes adiabatic ab initio electronic wave functions, Zhu and Yarkony J. Chem. Phys. 140, 024112 (2014). The algorithm uses derivative couplings in the defining equations for the diabatic Hamiltonian, H(d), and fits all its matrix elements simultaneously to adiabatic state data. This procedure ultimately provides an accurate, quantifiably diabatic, representation of the adiabatic electronic structure data. However, optimizing the large number of nonlinear parameters in the basis functions and adjusting the number and kind of basis functions from which the fit is built, which provide the essential flexibility, has proved challenging. In this work, we introduce a procedure that combines adiabatic state and diabatic state data to efficiently optimize the nonlinear parameters and basis function expansion. Further, we consider using direct properties based diabatizations to initialize the fitting procedure. To address this issue, we introduce a systematic method for eliminating the debilitating (diabolical) singularities in the defining equations of properties based diabatizations. We exploit the observation that if approximate diabatic data are available, the commonly used approach of fitting each matrix element of H(d) individually provides a starting point (seed) from which convergence of the full H(d) construction algorithm is rapid. The optimization of nonlinear parameters and basis functions and the elimination of debilitating singularities are, respectively, illustrated using the 1,2,3,4(1)A states of phenol and the 1,2(1)A states of NH3, states which are coupled by conical intersections.

Analysis of geometric phase effects in the quantum-classical Liouville formalism

We analyze two approaches to the quantum-classical Liouville (QCL) formalism that differ in the order of two operations: Wigner transformation and projection onto adiabatic electronic states. The analysis is carried out on a two-dimensional linear vibronic model where geometric phase (GP) effects arising from a conical intersection profoundly affect nuclear dynamics. We find that the Wigner-then-Adiabatic (WA) QCL approach captures GP effects, whereas the Adiabatic-then-Wigner (AW) QCL approach does not. Moreover, the Wigner transform in AW-QCL leads to an ill-defined Fourier transform of double-valued functions. The double-valued character of these functions stems from the nontrivial GP of adiabatic electronic states in the presence of a conical intersection. In contrast, WA-QCL avoids this issue by starting with the Wigner transform of single-valued quantities of the full problem. As a consequence, GP effects in WA-QCL can be associated with a dynamical term in the corresponding equation of motion. Since the WA-QCL approach uses solely the adiabatic potentials and non-adiabatic derivative couplings as an input, our results indicate that WA-QCL can capture GP effects in two-state crossing problems using first-principles electronic structure calculations without prior diabatization or introduction of explicit phase factors.

Full-dimensional potentials and state couplings and multidimensional tunneling calculations for the photodissociation of phenol

Full-dimensional potentials and state couplings were developed for the photodissociation of phenol. We also present multidimensional tunneling calculations at the transition state on the shoulder of the first conical intersection.

Determining partial differential cross sections for low-energy electron photodetachment involving conical intersections using the solution of a Lippmann-Schwinger equation constructed with standard electronic structure techniques

A method for obtaining partial differential cross sections for low energy electron photodetachment in which the electronic states of the residual molecule are strongly coupled by conical intersections is reported. The method is based on the iterative solution to a Lippmann-Schwinger equation, using a zeroth order Hamiltonian consisting of the bound nonadiabatically coupled residual molecule and a free electron. The solution to the Lippmann-Schwinger equation involves only standard electronic structure techniques and a standard three-dimensional free particle Green's function quadrature for which fast techniques exist. The transition dipole moment for electron photodetachment, is a sum of matrix elements each involving one nonorthogonal orbital obtained from the solution to the Lippmann-Schwinger equation. An expression for the electron photodetachment transition dipole matrix element in terms of Dyson orbitals, which does not make the usual orthogonality assumptions, is derived.

Space-time contours to treat intense field-dressed molecular states

In this article we consider a molecular system exposed to an intense short-pulsed external field. It is a continuation of a previous publication [A. K. Paul, S. Adhikari, D. Mukhopadhyay et al., J. Phys. Chem. A 113, 7331 (2009)] in which a theory is presented that treats quantum effects due to nonclassical photon states (known also as Fock states). Since these states became recently a subject of intense experimental efforts we thought that they can be treated properly within the existing quantum formulation of dynamical processes. This was achieved by incorporating them in the Born-Oppenheimer (BO) treatment with time-dependent coefficients. The extension of the BO treatment to include the Fock states results in a formidable enhancement in numerical efforts expressed, in particular, in a significant increase in CPU time. In the present article we discuss an approach that yields an efficient and reliable approximation with only negligible losses in accuracy. The approximation is tested in detail for the dissociation process of H(2) (+) as caused by a laser field.

Intralines of quasi-conical intersections on torsion planes: methylamine as a case study

Recently we reported on a novel feature associated with the intersection of the two lowest states (1)A' and (1)A'' of the methylamine (J. Chem. Phys. 2008, 128, 244302). We established the existence of a finite (closed) line of conical intersections (ci), namely, a finite seam, located in the HC-NHH symmetry plane, a line that is formed by moving a single hydrogen on that plane while locking the positions of the (six) other atoms. In the present article, this study is extended to the corresponding torsion planes formed by rotating the methyl group around the CN axis. The torsion planes, in contrast with the HC-NHH symmetry plane, do not satisfy the symmetry feature that enables the seam just mentioned. Nevertheless, the calculated nonadiabatic coupling terms (NACTs) resemble features similar to those encountered in the HC-NHH symmetry plane. Following a tedious numerical study supported by a theoretical model (Section III), it was verified that these NACTs may become similar to those on the symmetry plane, sometimes even to the level of almost no distinction, but lack one basic feature; namely, they are not singular and therefore do not form topological effects.

The simulated photoelectron spectrum of 1-propynide

The negative ion photoelectron spectrum of 1-propynide is computed by employing the multimode vibronic coupling approach. A three-state quasidiabatic Hamiltonian, H(d), is reported, which accurately represents the ab initio determined equilibrium geometries and harmonic frequencies of the ground X (2)A(1) state as well as the low-lying Jahn-Teller distorted components of the A (2)E excited state. It also reproduces both the minimum energy crossing point (MECP) on the symmetry-required (2)E(x)-(2)E(y) conical intersection seam and the MECP on the same symmetry (2)A(1)-(2)E(x) conical intersection seam. H(d) includes all terms through second order in internal coordinates for both the diagonal and off-diagonal blocks. It is centered at the (2)E(x)-(2)E(y) MECP and is determined using ab initio gradients and derivative couplings near both the (2)E(x)-(2)E(y) MECP and the X (2)A(1) equilibrium geometry. This construction is enabled by a recently reported normal equation based algorithm. The C(3v) symmetry of the system is used to significantly reduce the computational cost of the ab initio treatment. This H(d) is then expressed in a vibronic basis that is chosen for its ability to reduce the dimension of the vibronic expansion. The vibronic Hamiltonian matrix is diagonalized to obtain a negative ion photoelectron spectrum for 1-propynide-h(3). The determined spectrum compares favorably with previous spectroscopic results. In particular, the lines attributable to the (2)E state are found to be much weaker than those corresponding to the (2)A(1) state of 1-propynyl. This diminution of the (2)E state is attributable principally to the (2)E(x)-(2)A(1) conical intersection rather than an intrinsically small electronic transition moment for the production of the (2)E state.