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

Saxe, Paul; Lengsfield, Byron H.; Yarkony, David R.

doi:10.1016/0009-2614(85)80935-0

Cited by 110 publications

(82 citation statements)

References 38 publications

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“…In a different field, location and characteristics of ci's on diabatic potential surfaces have been recognized as essential for the evaluation of dynamic parameters, like nonadiabatic coupling terms, needed for the dynamic and static properties of some molecules ( [277] - [280], [192]). More recently, pairs of ci's have been studied [281,282] in greater detail.…”

Section: Non-linearities That Lead To Multiple Degeneraciesmentioning

confidence: 99%

Complex States of Simple Molecular Systems

Englman¹,

Yahalom²

2002

Advances in Chemical Physics

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A review is given of phase properties in molecular wave functions, composed of a number of (and, at least, two) electronic states that become degenerate at some nearby values of the nuclear configuration. Apart from discussing phases and interference in classical (non-quantal) systems, including light-waves, the review looks at the constructability of complex wave functions from observable quantities ("the phase problem"), at the controversy regarding quantum mechanical phase-operators, at the modes of observability of phase and at the role of phases in some non-demolition measurements. Advances in experimental and (especially) theoretical aspects of Aharonov-Bohm and topological (Berry) phases are described, including those involving two-electron and relativistic systems. Several works in the phase control and revivals of molecular wave-packets are cited as developments and applications of complex-function theory. Further topics that this review touches on are: coherent states, semiclassical approximations and the Maslov index. The interrelation between time and the complex state is noted in the contexts of time delays in scattering, of time-reversal invariance and of the existence of a molecular time-arrow.When the stationary Born-Oppenheimer description for nearly degenerate state is regarded as "embedded" in a broader dynamic formulation, namely through solution of the time dependent Schrödinger equation, the wave function becomes necessarily complex. The analytic behavior of this function in the complex time-plane can be exploited to gain information about the connection between phases and moduli of the componentamplitudes. One form of this connection are a pair of reciprocal (Kramers-Kronig type) relations in the complex time-domain. These show that certain phase changes in a component amplitude (such as, e.g., lead to a non-zero Berry-phase) require changes in the amplitude-moduli, too, and imply degeneracies of the electronic state.The subject of conical degeneracy, or intersection, of adjacent potential surface is extended to nonlinear nuclear-electronic coupling, which can then result in multiple conical degeneracies on a nuclear coordinate plane. We treat cases of double and fourfold intersections, the latter under trigonal symmetry, and employ an analytic-graphical phase tracing method to obtain the resulting Berry phases as the system circles around some or all of the degeneracy points. These phases can take up values that are all integral multiples of π, with integers that vary with the physical situation (or with the model postulated for it).A further type of invariance, with respect to gauge transformation, is also tied to the complex form of the wave function. For a many-component state this leads to a consideration of tensorial (Yang-Mills type) fields F for molecular systems. It is shown that it is the truncation of the Born-Oppenheimer electron-nuclear superposition that generates the molecular Yang-Mills fields.The equations of motion for the nuclear degrees of freedom are derived from a Lagr...

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Section: Non-linearities That Lead To Multiple Degeneraciesmentioning

confidence: 99%

Complex States of Simple Molecular Systems

Englman¹,

Yahalom²

2002

Advances in Chemical Physics

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“…In addition, I is a 2ϫ2 identity matrix, (q) are respectively, the first-derivative [25][26][27][28] and second-derivative 25,29,30 nonadiabatic coupling matrix elements between the ground and firstexcited adiabatic electronic PESs. For the two-electronicstate approximation they are 2ϫ2 matrices, whose elements are defined by…”

Section: Iϩ2wmentioning

confidence: 99%

Accurate first-derivative nonadiabatic couplings for the H3 system

Abrol

Shaw

Kuppermann

et al. 2001

The Journal of Chemical Physics

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A conical intersection exists between the ground ͑1 2 AЈ) and the first-excited ͑2 2 AЈ) electronic potential energy surfaces ͑PESs͒ of the H 3 system for C 3v geometries. This intersection induces a geometric phase effect, an important factor in accurate quantum mechanical reactive scattering calculations, which at low energies can be performed using the ground PES only, together with appropriate nuclear motion boundary conditions. At higher energies, however, such calculations require the inclusion of both the 1 2 AЈ and 2 2 AЈ electronic PESs and the corresponding nuclear derivative couplings. Here we present ab initio first-derivative couplings for these states obtained by analytic gradient techniques and a fit to these results. We also present a fit to the corresponding 1 2 AЈ and 2 2 AЈ adiabatic electronic PESs, obtained from the ab initio electronic energies. The first-derivative couplings are compared with their approximate analytical counterparts obtained by Varandas et al. ͓J. Chem. Phys. 86, 6258 ͑1987͔͒ using the double many-body expansion method. As expected, the latter are accurate close to conical intersection configurations but not elsewhere. We also present the contour integrals of the ab initio couplings along closed loops around the above-mentioned conical intersection, which contain information about possible interactions between the 2 2 AЈ and 3 2 AЈ states.

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“…The representation of their behavior by means of simple analytic models, as has been done in Section 2, facilitates theoretical treatment. Thus, for these systems an immediate application of this work is in the evaluation of dynamic parameters, like nonadiabatic coupling terms, needed for the dynamic and static properties of some molecules [17,18,21,22]. Some emphasis has been put in this work on the development of the phase during a slow nuclear motion in the vicinity of multiple CIs.…”

Section: Discussionmentioning

confidence: 99%

Experimental and calculational consequences of phases in molecules with multiple conical intersections

Englman

Yahalom

Baer³

et al. 2003

Int J of Quantum Chemistry

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ABSTRACT:We set up a theoretical model for treating several degeneracies between two adjacent potential energy surfaces in molecular systems, e.g., cases of four (or more) conically intersecting degeneracies, located in a plane formed by two-fold molecular displacement coordinates, in trigonal (or cubic) symmetry and twin conical intersections (CIs) for molecules with two-fold symmetries. When the system circles (in a time-variant manner) entirely inside or entirely outside these CIs, it picks up phases (the geometric phases) that are zero or 2N. Here, N is (to a good approximation) an integer whose value depends on the Hamiltonian for the system, on the distance of circling from the CI, and on the speed of circling. For the adiabatic (slow circling) limit, we develop a simple algebraic-graphical method that yields the value of N in each case. Even in this limit, the value of N is model dependent and is not uniquely given by the number of CIs that are circled. Further, we suggest an experiment (based on a pumpprobe method) for tracing the time development of a system that is subject to a periodically varying field carrying it around the CI. The experiment leads to relations between the spectroscopic transition strengths and state amplitudes, such that the (open-path) phases are fully given (including the additive 2N term and excluding the dynamic phase). Finally, we compare geometric phases that differ by integral multiples of 2 and, therefore, cannot be distinguished by phase interference. We show that the different cases have, nevertheless, observational correlates in the number of times that zero or unity occurs in a wave function component amplitude during circling. The rules of correspondence are given.

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

Cited by 110 publications

References 38 publications

Complex States of Simple Molecular Systems

Complex States of Simple Molecular Systems

Accurate first-derivative nonadiabatic couplings for the H3 system

Experimental and calculational consequences of phases in molecules with multiple conical intersections

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