David R. Yarkony

Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. I. Formalism

An efficient and general method for the analytic computation of the nonandiabatic coupling vector at the multireference configuration interaction (MR-CI) level is presented. This method is based on a previously developed formalism for analytic MR-CI gradients adapted to the use for the computation of nonadiabatic coupling terms. As was the case for the analytic energy gradients, very general, separate choices of invariant orbital subspaces at the multiconfiguration self-consistent field and MR-CI levels are possible, allowing flexible selections of MR-CI wave functions. The computational cost for the calculation of the nonadiabatic coupling vector at the MR-CI level is far below the cost for the energy calculation. In this paper the formalism of the method is presented and in the following paper [Dallos et al., J. Chem. Phys. 120, 7330 (2004)] applications concerning the optimization of minima on the crossing seam are described.

Role of Conical Intersections in Molecular Spectroscopy and Photoinduced Chemical Dynamics

This review describes how conical intersections affect measured molecular spectra and simple photofragmentation processes. We consider excitations that result in electron ejection, that is, photoionization or photodetachment, as well as photoinduced H-atom elimination. Section 1 presents a brief overview of the history of conical intersections and their rise from an arcane theoretical concept to a major paradigm in nonadiabatic chemistry. In Section 2, the generic properties of conical intersections are discussed, as well as their characterization with modern electronic-structure methods. Section 3 briefly discusses computational tools used to compute the nuclear motion involving conical intersections. Section 4 describes how the ideas of Sections 2 and 3 are combined to simulate molecular spectra impacted by conical intersections. Section 5 describes selected recent experimental and computational studies of photoelectron, photodetachment, and photofragment spectra. Rather than providing an encyclopedic bibliography of the previous and current literature, we illustrate significant problems currently being addressed and describe what can be accomplished with current computational techniques and how these results are achieved. Section 6 suggests future directions in this field.

Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. II. Minima on the crossing seam: Formaldehyde and the photodimerization of ethylene

The method for the analytic calculation of the nonadiabatic coupling vector at the multireference configuration-interaction (MR-CI) level and its program implementation into the COLUMBUS program system described in the preceding paper [Lischka et al., J. Chem. Phys. 120, 7322 (2004)] has been combined with automatic searches for minima on the crossing seam (MXS). Based on a perturbative description of the vicinity of a conical intersection, a Lagrange formalism for the determination of MXS has been derived. Geometry optimization by direct inversion in the iterative subspace extrapolation is used to improve the convergence properties of the corresponding Newton-Raphson procedure. Three examples have been investigated: the crossing between the 1(1)B1/2(1)A1 valence states in formaldehyde, the crossing between the 2(1)A1/3(1)A1 pi-pi* valence and ny-3py Rydberg states in formaldehyde, and three crossings in the case of the photodimerization of ethylene. The methods developed allow MXS searches of significantly larger systems at the MR-CI level than have been possible before and significantly more accurate calculations as compared to previous complete-active space self-consistent field approaches.

Nonadiabatic quantum chemistry--past, present, and future.

1. Background and Definitions 481 1.1. Some History 481 1.2. Conical Intersections, Derivative Couplings and the Diabatic Representation 482 1.3. Classifying Conical Intersections: Symmetry Considerations 483 1.4. Prevalence of Conical Intersections 483 1.5. Locating and Characterizing Conical Intersections 483 2. Current State of the Art 484 2.1. Conical Intersections and Radiationless Decay 484 2.1.1. Radiationless Decay of Furan (C4H4O) 484 2.2. Diabatic States and the Representation of Adiabatic Potential Energy Surfaces and Their Couplings 485 2.2.1. H for Bound States 486 2.2.2. H for Dissociative States 486 2.2.3. Determining H 486 2.3. Nuclear Dynamics for Electronically Nonadiabatic Processes 487 2.4. Nonadiabatic Effects near Surfaces and Interfaces 487 2.4.1. Semiconductor Interfaces 487 2.4.2. Metal Surfaces 487 2.5. Effects of the Environment on Nonadiabatic Processes 488 2.6. Control of Nonadiabatic Chemical Dynamics with Lasers 489 2.6.1. Routing and the Branching Plane 489 2.6.2. Controlling Photoisomerization 489 2.6.3. Cyclohexadiene (CHD) Hexatriene (HT) Photoisomerization 490 2.6.4. Vibrationally Mediated Photodissociation of NH3 490 2.6.5. Controlling Radiationless Decay 491 2.7. Conical Intersections and Mechanism for Nonadiabatic Reactions in Photobiology 491 2.7.1. Electron-Driven Proton Transfer 491 2.7.2. Intersection Space, ReactionMechanisms, and Nonadiabatic Dynamics 491 2.8. Nonadiabatic Photoelectron Spectroscopy 491 2.8.1. General Formulation Vibronic Coupling Model 492

INTERACTION POTENTIAL BETWEEN TWO RIGID HF MOLECULES

As a prelude to the study of energy transfer in the HF–HF system, the potential energy surface for the interaction of two rigid HF molecules has been calculated within the ab initio self‐consistent‐field framework. An H(4s 1p/2s 1p), F(9s 5p 1d/4s 2p 1d) basis set of contracted Gaussian function was employed. The number of unique points on the surface is greatly reduced by symmetry, and only 294 points were required to give a fairly complete description of the four‐dimensional surface. Parts of the surface are illustrated by a series of contour maps. Some preliminary attempts to fit the surface to an analytic form are described. The equilibrium geometry of (HF)2 is predicted.

On the evaluation of nonadiabatic coupling matrix elements using SA‐MCSCF/CI wave functions and analytic gradient methods. I

A method for the efficient evaluation of nonadiabatic coupling matrix elements of the form 〈ΨI‖∂/∂RαΨJ〉 is presented. The wave functions ΨI and ΨJ are assumed to be multiconfiguration self‐consistent field (MCSCF) wave functions optimized within the state averaged approximation. The method, which can treat several states simultaneously, derives its efficiency from the direct solution of the coupled perturbed state averaged MCSCF equations and the availability of other appropriate derivative integrals. An extension of this approach to SA‐MCSCF/CI wave functions is described. Here it is shown that computational efficiencies can be achieved by exploiting analogies with analytic CI gradient methods. Numerical examples for C2v approach of Mg to H2 are presented.

Nuclear dynamics near conical intersections in the adiabatic representation: I. The effects of local topography on interstate transitions

The local topography of a conical intersection can be represented by four parameters, readily determined from multireference configuration interaction wave functions, describing the pitch and tilt of the double cone. The time-dependent Schrodinger equation is solved in the vicinity of a conical intersection in the adiabatic basis using an approach tailored to this representation. It is shown that an adiabatic state treatment, which offers conceptual advantages is, in the appropriate set of internal coordinates, not qualitatively more difficult than the equivalent calculation in a diabatic basis. The present treatment is fully hermitian and takes full account of the geometric phase effect being, for example, gauge invariant (in the infinite basis limit) and could be used to develop a fully adiabatic description of nonadiabatic dynamics. The gauge invariant formulation provides interesting insights into the consequences of neglecting the geometric phase. The algorithm is used to study the effects of the dou...

On the intersection of two potential energy surfaces of the same symmetry. Systematic characterization using a Lagrange multiplier constrained procedure

Two nonrelativistic Born–Oppenheimer potential energy surfaces of the same space‐spin symmetry may intersect on a surface of dimension N−2, where N is the number of internal nuclear degrees of freedom. Characterization of this entire surface can be quite costly. An algorithm, employing multiconfiguration self‐consistent‐field (MCSCF)/configuration interaction(CI) wave functions and analytic gradient techniques, is presented that avoids the determination of the full N−2 dimensional surface, while directly locating portions of the crossing surface that are energetically important. The algorithm determines extrema of the Lagrangian function LIJ(R,ξ,λ) = EI(R) + ξ1[EI(R) − EJ(R)] + ξ2HIJ(R)/2+ ∑Mk=1λkCk(R), where Ck(R) is any geometric equality constraint such as RKL2−αKL2=0, or RKL2−RMN2=0, RKL=‖RK−RL‖ and the ξ and λ are Lagrange multipliers. The efficacy of this algorithm is demonstrated using a MCSCF/first order CI description of 1,22A’ states of HCO.

On the consequences of nonremovable derivative couplings. I. The geometric phase and quasidiabatic states: A numerical study

Conical intersections complicate the computational treatment of nuclear dynamics in the adiabatic state basis through the geometric phase effect and singularities in the derivative couplings. The diabatic representation seeks to eliminate these difficulties. However, the adiabatic to diabatic state transformation is necessarily approximate in a polyatomic molecule since the derivative couplings cannot be rigorously removed. This point is rarely considered when constructing approximate diabatic states. The nonremovable part of the derivative couplings is investigated by considering the integral of the derivative coupling along closed loops in the vicinity of the 1 2A′–2 2A′ seam of conical intersections in H3.

On the evaluation of non-adiabatic coupling matrix elements for large scale CI wavefunctions

Abstract The implementation of a recently proposed technique for evaluating matrix elements of the form 〈Ψ J ( r ; R )|∂Ψ I ( r ; R )/t6 R α 〉 r using analytic gradient techniques is described. The Ψ K ( r ; R ) are developed from state-averaged multiconfiguration self-consistent-field/configuration interaction (CI) wavefunctions. The CI wavefunctions are determined using the shape driven graphical unitary group approach. The method is shown to be considerably more efficient than presently existing approaches based on divided differences. As an illustration of the potentialities of this approach non-adiabatic coupling matrix elements are determined for the collinear charge transfer reaction: Mg( 1 S) + FH( 1 Σ + )→MgF( 2 Σ + )+H( 2 S).