Toward the Full Quantum Dynamical Description of Photon Induced Processes in D2.

The dissociative ionization (multiphoton regime) of the D2+ ion by ultrashort laser pulses has been studied theoretically using ab initio calculations. The combined ionization and dissociation spectrum was explored for fixed molecular axis orientations. In accordance with previous investigations, the dominant features in the obtained joint energy spectrum were multiphoton peaks. In addition to this, in the present work, photoelectron angular distributions were analyzed as well. By performing a partial wave analysis for each multiphoton peak, we have identified the number of absorbed photons. Moreover, we also found that the angular distribution can significantly change inside a multiphoton peak as a function of electron and nuclear kinetic energy.

Direct Signature of Light-Induced Conical Intersections in Diatomics.

Nonadiabatic effects are ubiquitous in physics, chemistry, and biology. They are strongly amplified by conical intersections (CIs), which are degeneracies between electronic states of triatomic or larger molecules. A few years ago it was revealed that CIs in molecular systems can be formed by laser light, even in diatomics. Because of the prevailing strong nonadiabatic couplings, the existence of such laser-induced conical intersections (LICIs) may considerably change the dynamical behavior of molecular systems. By analyzing the photodissociation process of the D2+ molecule carefully, we found a robust effect in the angular distribution of the photofragments that serves as a direct signature of the LICI, providing undoubted evidence of its existence.

Dressed adiabatic and diabatic potentials to study topological effects for F + H2.

We report here on the two lowest, rigorous-accurate diabatic potential energy surfaces (PES), for the F + H2 system, as calculated by including the two dominant topological effects of this system at the low energy region, namely, the Jahn-Teller effect and the Renner-Teller effect. Both effects were treated in the most rigorous way as demanded by the Born-Oppenheimer approach. No approximations were made, and in those cases where convergence was required, it was satisfied. In other words, convergence was attained in all situations. The numerical part that includes the calculation of the two lowest ab initio adiabatic PESs and the corresponding nonadiabatic coupling terms (NACTs) was carried out using the MOLPRO program. The required diabatic potentials are calculated by employing these ab initio adiabatic PESs and the corresponding adiabatic-to-diabatic angles as obtained employing the above-mentioned ab initio NACTs. The relevance of these Renner-Teller/Jahn-Teller diabatic potentials is studied by comparing the dressed-lowest ab initio adiabatic PES and the one formed by diagonalizing the dressed-diabatic 2 × 2 potential matrix. The dressed-potentials are calculated employing the vib-rotational manifold derived for each of the three surfaces, namely, the lowest adiabatic potential and the two diabatic ones. This kind of study was recently recommended by Lipoff and Herschbach ( Mol. Phys. 2010 , 108 , 1133 ) as a "blessed-practice" for the relevance of any PES. In the present case significant differences were revealed between the two types of dressed-adiabatic PESs, eventually, indicating that the lowest, ab initio PES (due to the Born-Oppenheimer approximation) is not adequate for low energy processes.

Dressed adiabatic and diabatic potentials for the Renner-Teller/Jahn-Teller F + H2 system.

We follow a suggestion by Lipoff and Herschbach (Mol. Phys. 2010, 108, 1133) and compare dressed potentials to get insight regarding the low-energy dynamics (e.g., cold reaction) taking place in molecular systems. In this particular case we are interested in studying the effect of topological effects on the interacting atoms. For this purpose we consider dressed adiabatic and adiabatic-via-dressed diabatic potentials in the entrance channel of reactive systems. In a recent study of this kind for the F + H2 system (J. Chem. Phys. 2012, 136, 054104), we revealed that a single Jahn-Teller conical intersection is expected to have only a mild effect on the dynamics. This fact implies that the Born-Oppenheimer approximation is expected to be valid for this system at least for low enough energies. In the present article this study is extended to include also the Renner-Teller effect as produced by the two lower degenerate Π states. As a result we consider three electronic states which enforce the use of the adiabatic-to-diabatic transformation (ADT) matrix A. The results indicate that the topological effects as produced by the extended Renner/Teller-Jahn/Teller system are strong to the level that, most likely, abolishes the Born-Oppenheimer approximation for this system, all this in contrast to our previous findings (see above publication).

A tri-atomic Renner-Teller system entangled with Jahn-Teller conical intersections.

The present study concentrates on a situation where a Renner-Teller (RT) system is entangled with Jahn-Teller (JT) conical intersections. Studies of this type were performed in the past for contours that surround the RT seam located along the collinear axis [see, for instance, G. J. Halász, Á. Vibók, R. Baer, and M. Baer, J. Chem. Phys. 125, 094102 (2006)]. The present study is characterized by planar contours that intersect the collinear axis, thus, forming a unique type of RT-non-adiabatic coupling terms (NACT) expressed in terms of Dirac-δ functions. Consequently, to calculate the required adiabatic-to-diabatic (mixing) angles, a new approach is developed. During this study we revealed the existence of a novel molecular parameter, η, which yields the coupling between the RT and the JT NACTs. This parameter was found to be a pure number η = 22/π (and therefore independent of any particular molecular system) and is designated as Renner-Jahn coupling parameter. The present study also reveals an unexpected result of the following kind: It is well known that each (complete) group of states, responsible for either the JT-effect or the RT-effect, forms a Hilbert space of its own. However, the entanglement between these two effects forms a third effect, namely, the RT/JT effect and the states that take part in it form a different Hilbert space.

Quantum dynamics through conical intersections: combining effective modes and quadratic couplings.

We present a detailed study for the short-time dynamics through conical intersections in molecular systems related to the quadratic vibronic coupling (QVC) Hamiltonian [Müller, H.; Köppel, H.; Cederbaum, L. S. New J. Chem. 1993, 17, 7-29] and the effective-mode formalism [Cederbaum, L. S.; Gindensperger, E.; Burghardt, I. Phys. Rev. Lett. 2005, 94, 113003]. Our approach is based on splitting the nuclear degrees of freedom of the whole system into system modes and environment modes. It was found that only three-effective environmental modes together with the system's modes are needed to describe the short-time dynamics of the complex system correctly. In addition, a detailed mathematical proof is given in the appendix to demonstrate that the exact cumulants are recovered up to the second order within the cumulant expansion of the autocorrelation function. The butatriene molecule is studied as an explicit showcase example to stress the viability of our proposed scheme and to compare with other systems.

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.

An intraline of conical intersections for methylamine.

In this article are considered the conical intersections (ci's) related to the N-H bond in the methylamine, CH(3)NH(2), molecule. The novel feature that was revealed is that the two lowest states 1A(') and 1A(") are coupled by a line of cis located in HC-NHH plane-a line that is formed by moving a single hydrogen on that plane while fixing the (six) other atoms. The validity of this line was proven first by studying the singularities of the (angular) nonadiabatic coupling terms and then by revealing the degeneracy points formed by the two interacting adiabatic potential energy surfaces (PESs). A theoretical analysis indicated that the line has to be a finite closed line. We also calculated the Berry phase for a contour that surrounds this line and found it to be 3.127 rad, namely, a value reasonably close to pi. The existence of such lines of cis-instead of isolated cis (as exhibited by other n-atomic (n>3) molecules such as HNCO or C(2)H(2))-may enhance significantly the transition rate from an upper adiabatic state to a lower one. There are also numerical advantages in such situations, that is, if such a line is properly placed in that plane (like in the present case) the wave-packet treatment of the nuclei can be carried out employing a single diabatic PES instead of having to consider two coupled PESs.

Topological effects for nonsymmetrical configurations: the C2H+2 as a case study.

During the last decade the study of topological effects formed by molecular systems became a routine but it was always carried out for configurations that were limited by symmetry conditions. To be more specific this applied to the Jahn-Teller (JT) effect formed by molecular configurations of planar symmetry [see, e.g., Baer et al., Faraday Discuss. 127, 337 (2004)] and the Renner-Teller effect formed by configurations of axial symmetry [see, e.g., Halasz et al., J. Chem. Phys. 126, 154309 (2007)]. In this article we consider for the first time molecular configurations that avoid any symmetry conditions or, in other words, are characterized by the C(1) point group. We report on a detailed study of topological effects formed by such a molecular system. The study concentrates on both, the two-state (Abelian) case and the multistate (non-Abelian) case. It is shown that the theory that was originally developed to treat topological effects due the JT intersection and also applies for the study of topological effects in the most general case. The study is accompanied with numerical results.

Renner-Teller/Jahn-Teller intersections along the collinear axes of polyatomic molecules: C2H2 + as a case study.

Recently we discussed the Renner-Teller effect in triatomic molecules [J. Chem. Phys. 125, 094102 (2006)]. In that article the main message is that the Renner-Teller phenomenon, just like the Jahn-Teller phenomenon, is a topological effect. Now we extend this study to a tetra-atomic system, namely, the C(2)H(2) (+) ion, for which topological effects are revealed when one atom surrounds the triatom axis or when two atoms surround (at a time) the two-atom axis. The present study not only supports the findings of the previous study, in particular, the crucial role played by the topological D matrix for diabatization, but it also reveals new features which are expected to be more and more pronounced the larger the original collinear molecule. As already implied, shifting away two atoms from the collinear molecular axis does not necessarily abolish the ability of the remaining two atoms to form topological effects. Moreover, the study indicates that when the two hydrogens are shifted away, the CC axis produces two kinds of topological effects: (1) a Renner-Teller effect (characterized by a topological phase of 2pi) which is revealed when the two hydrogens surround, rigidly, this axis (as mentioned above), and (2) a Jahn-Teller effect (characterized by a topological phase of pi) which is revealed when one of the hydrogens surrounds this axis while the other hydrogen is clamped to its position.

The electronic nonadiabatic coupling term: can it be ignored in dynamic calculations?

Whereas the search for the degeneracy points which are better known as conical intersections (or ci-points) is usually carried out with a lot of devotion, the nonadiabatic coupling terms (NACTs) which together with the adiabatic potential energy surfaces appear in the nuclear Born-Oppenheimer-Schrodinger equation are ignored in most dynamical calculations. In the present article we consider two well known frameworks, namely, the semiclassical surface hopping method and the vibrational coupling model Hamiltonian that avoid the NACTs and examine to what extent, this procedure is justified.

D matrix analysis of the Renner-Teller effect: an accurate three-state diabatization for NH2.

Some time ago we published our first article on the Renner-Teller (RT) model to treat the electronic interaction for a triatomic molecule [J. Chem. Phys. 124, 081106 (2006)]. The main purpose of that Communication was to suggest considering the RT phenomenon as a topological effect, just like the Jahn-Teller phenomenon. However, whereas in the first publication we just summarized a few basic features to support that idea, here in the present article, we extend the topological approach and show that all the expected features that characterize a three (multi) state RT-type'3 system of a triatomic molecule can be studied and analyzed within the framework of that approach. This, among other things, enables us to employ the topological D matrix [Phys. Rev. A 62, 032506 (2000)] to determine, a priori, under what conditions a three-state system can be diabatized. The theoretical presentation is accompanied by a detailed numerical study as carried out for the HNH system. The D-matrix analysis shows that the two original electronic states 2A1 and 2B1 (evolving from the collinear degenerate Pi doublet), frequently used to study this Renner-Teller-type system, are insufficient for diabatization. This is true, in particular, for the stable ground-state configurations of the HNH molecule. However, by including just one additional electronic state--a B state (originating from a collinear Sigma state)--it is found that a rigorous, meaningful three-state diabatization can be carried out for large regions of configuration space, particularly for those, near the stable configuration of NH2. This opens the way for an accurate study of this important molecule even where the electronic angular momentum deviates significantly from an integer value.

Renner-Teller nonadiabatic coupling terms: An ab-initio study of the HNH molecule.

In this Communication we present the first theoretical/numerical treatment of nonadiabatic coupling terms (NACT) that originate from the Renner-Teller (RT) model, namely, those that follow from the splitting of an electronic level of a linear molecule when it becomes bent. These two newly formed states are characterized by different symmetries and are designated as A and B. Our main findings: (1) The RT NACTs are quantized as long as they are calculated close enough to collinear configuration of the molecule (in this case HNH). Their value is tau = 1 (the Jahn-Teller values in similar situations, are tau = (1/2)). (2) Calculation of RT NACTs at bent configurations (i.e., at a distance from the linear axis) yield decreased values, sometimes by more than 50%. This last finding implies that in strongly bent configurations the two-state Hilbert subspace (formed by the above mentioned A and B states) is affected by upper states, most likely via Jahn-Teller conical intersections. (3) This study has also important practical implications. The fact that the RT NACTs decrease in (strongly) bent situations implies that analyzing spectroscopic data employing only the two Pi-states may not be sufficient in order to achieve the required accuracy.

Two-state versus three-state quantization: an ab initio study of the three lower states of the {N,H2/A'} system.

In this article we present the first ab initio study of the conical intersections (cis) and their electronic nonadiabatic coupling terms (NACTs) for the {N,H2} system. Efforts were made to reveal the location of cis between the two lower, 1 2A' and 2 2A' states--to be designated as (1,2) cis--and the cis between the two upper, 2 2A' and 3 2A' states--to be designated as the (2,3) cis--of this system. We found that these cis are located along the collinear {NHH) arrangement. The study is carried out by analyzing two-state magnitudes such as the (1,2) and (2,3) adiabatic-to-diabatic transformation angles (known also as the mixing angles) and the corresponding topological phases (known also as the Berry phases or the Longuet-Higgins phases). In addition, a detailed three-state study is carried out. Here the emphasis is on driving the diagonal elements of the topological D matrix and analyzing situations for which the corresponding nonadiabatic coupling matrix is quantized. The reliability of two-state results is carefully examined by comparing them with corresponding outcomes derived for the three-state study. In addition we also calculated the potential-energy surfaces related to the two lower states and studied to what extent they are affected by the (1,2) ci. The results obtained in this treatment were found to be in full agreement with the NACT's calculations.

N-state adiabatic-to-diabatic transformation angle: theory and application.

In this article is discussed a new diabatization procedure which is expected to be reliable and, also, relatively easy to implement. This procedure takes into account the two main ingredients related to diabatization: (1) The size N of the smallest (relevant) group of states that forms a Hilbert subspace (this fact enforces the dimension of the adiabatic-to-diabatic transformation matrix to be N). (2) The total energy E which determines the number of open states, p, within this group of N states. The main emphasis in this manuscript is on the case that N is arbitrary but p is equal to 2. The various derivations as well as the final results are accompanied by numerical examples extracted from three- to five-state ab initio calculations for the H + H2 system.

On diabatization and the topological D-matrix: theory and numerical studies of the H + H2 system and the C2H2 molecule.

This article is divided into two main parts: (1) The theoretical part contains a new derivation of the topological matrix D (M. Baer and A. Alijah, Chem. Phys. Lett., 2000, 319, 489) which is based, solely, on the spatial dependent electronic manifold. This derivation enables more intimate relations between the adiabatic and the diabatic frameworks as is discussed in detail in the manuscript. (2) The numerical part is also divided into two parts: (a) In the first part we extend our previous study on the H + H2 system (G. Halasz, A. Vibok, A. M. Mebel and M. Baer, J. Chem. Phys., 2003, 118, 3052) by calculating the topological matrix for five states (instead of three) and for configuration spaces four times larger than before. These studies are expected to yield detailed information on the possibility of diabatization of this system. (b) We report on preliminary results concerning the C2H2 molecule. So far we established the existence of one (1,2) conical intersection and we have good reasons to believe that this system contains several (2,3) and (3,4) conical intersections as well.

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.

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.

On the peculiarities of the diabatic framework: new insight.

In this article we consider the electronic diabatic presentation of a two-state system with the aim of earning insight regarding the distribution of conical intersections in a given region. In this process we revealed explicit relationship between the diabatic potentials and the locations of conical intersections. The study is accompanied with numerical examples as worked out for a model and ab initio potential energy surfaces of the Na+H2 system.