Rehovot, May 2007
See my recent talk on The Born-Oppenheimer Approach and Molecular Fields.
I. Topological Effects in Molecular Systems
I.1 First period: 1974-1990
In 1973, after my return from a two-year stay in the US, I decided to concentrate on the study of (quantum mechanical) charge transfer (CT) processes taking place during interactions between molecules and atomic ions. I familiarized myself with the Born-Oppenheimer (BO) treatment of molecular systems (as well as with the resulting adiabatic potential energy surfaces and the non-adiabatic coupling terms (NACTs)) and with an important paper by Felix Smith which elaborates on diabatic potentials for atom-atom systems. During summer 1974, I wrote what I consider to be my most innovative article (Chem. Phys. Lett. 35,112 (1975), Ref. 34), in which I discussed for the first time a transformation from the BO adiabatic framework to Lichten's diabatic framework for molecular systems. In what follows this transformation is termed the adiabatic-to-diabatic transformation, known also by the acronym ADT and will be labeled as A(s) where s stands for the relevant set of nuclear coordinates. In that paper I present the ADT matrix as a solution of first order differential equations to be solved along given contours in configuration space. Thus:
where t(s) is an anti-symmetric vector matrix that contains the (vectorial) non-adiabatic coupling terms. This equation can also be written as an integral equation along a given contour, G:
where A(s0) is the boundary condition for solving this equation. Eq. (1) has a solution given in terms of an exponentiated line integral (Molec. Phys. 40, 1011 (1980); Ref. 64):
where is the ordering operator and the dot stands for a scalar product. The exponentiated line integral is also known by the name: phase factor. It is important to mention that A(s) is an orthogonal (unitary) matrix
To complete the presentation we mention that in case of two states the ADT matrix can be presented in terms of an ADT angle (known also as the mixing angle) which is given in the form:
where t12(s|G) is the NACT between state 1 and state 2.
In that article is also discussed the so called Curl Condition which guarantees an analytic solution for the ADT matrix. The Curl-Condition is in fact the Curl expression for non-Abelian magnitudes (and is reminiscent of the Yang-Mills field which plays an important role in elementary particle physics).
where p and q are Cartesian coordinates.
Two comments have to be said with regard to article I: (i) This paper was sent out for publication on December 1974 but the refereeing process lasted about six months as it was rejected two-three times. (ii) About that time K.G. Wilson published a paper (Phys. Rev. D 10, 2445 (1974)) that in some respect is similar to this one but is associated with elementary particle physics.
The ideas expressed in the above-mentioned CPL paper served as the basis for everything I did later within the field of electronic non-adiabatic effects in molecular systems For instance the ADT matrix, derived in that paper, enables the elimination of the troublesome, singular NACTs, as well as the spiky conical intersections, from the BO adiabatic nuclear equation and yields a smooth, friendly and robust nuclear diabatic Schrödinger equation (SE). At this stage we mention the diabatic (full) potential matrix W(s) which is related to the BO adiabatic (diagonal) matrix u(s) as follows:
The resulting diabatic SE was solved on
various occasions but here I mention two numerical treatments: (1) The CT study
and (2) the study of CT and spin-transition for: Ar+H2+«Ar++H2
and exchange for (Ar+H2+;Ar++H2)®ArH++
H. In these two studies the results are compared with experiment: for
In addition I would like to mention the two
studies: (1) The QM treatment of exchange and CT processes for the four-center
I.2 Second period: 1992-2007
In 1984 Berry published his seminal paper on what is now known as the Berry phase. At a certain stage I wondered whether the (molecular) Berry phase is not somehow related to the two-state ADT angle (see Eq. (4)). For this purpose I started a fruitful and long standing collaboration with Professor Englman. In our first publication (Molec. Phys., 75, 293 (1992); Ref. 176), we suggested (following the analysis of the Jahn-Teller model) that the Berry Phase and the ADT angle, a(G) calculated along a closed-contour G:
are identical. This result is interesting because it implies that the Berry phase has its origin in the NACTs and therefore can be obtained from ab initio calculations for any desired closed contour. The importance of Eq. (7) is that it yields information on the number and position of the degeneracy points (also known as conical intersections or ci-points) between states 1 and 2. In what follows a(G) is termed as the topological phase.
A few years later this finding led to a much more general result, namely the introduction of the N-dimensional topological D-matrix which following Eq. (3) takes the form:
The main feature that characterizes Eq. (8) is that if a group of N-states forms, a Hilbert sub-space in a given nuclear region then the D-matrix has to be diagonal for every closed contour G in that region (Farad. Discuss. 127, 337 (2004); Ref. 292). In case the eigenfunctions that (that form t(s)) are real functions, the diagonal of the D-matrix contains ±1's. The number of -1's and their positions along the diagonal delivers information on the degeneracy points formed by the different (usually consecutive) pairs of states (Chem. Phys. Lett. 319, 480 (2000); Ref. 250; Phys. Rev. A, 62, 032506 (2000); Ref. 249 ).
In case the two states 1 and 2 (mentioned previously) form, in a given region, a two-state Hilbert space then a(G) as defined in Eq. (6) becomes an integer multiple of p. This seems to be another application to the Bohr-Sommerfeld quantization rule (Ref. 250).
The D-matrix not only yields information on the distribution of degeneracy points in a given region but also tells us if in the given region the relevant diabatic potential matrix (as defined in Eq. (6)) is single-valued.
Recently this theory was extended to treat molecular systems exposed to external fields (J. Phys. Chem. 107, 4724 (2003); Ref. 280; J. Chem. Phys. 119, 6998 (2003); Ref. 285). This extension leads to two interesting findings:
(i) The field-dressed NACT:
where the w-matrix is the Field-free « Field dressed transformation matrix.
(ii) The field dressed ADT matrix, and the corresponding space-time contours. This ADT matrix is a solution of two first order differential equations
to be solved along space-time contours (J. Phys. Chem. 110, 6571 (2006); Ref. 302; J. Chem. Phys. 126, 014106 (2007); Ref. 307).
The solution of Eqs. (10) can be presented as an exponential line-integral along the space-time contour Gst.
where M is the number of (s,t)-segments along the space-time contour Gst.
We are now in the process of employing this space-time contour for developing new approaches to treat molecular systems which are affected by intense external fields.
In what follows we mention briefly a few more achievements within this approach:
(1) We established the topological concept that says that the various phenomena related to the Longuet-Higgins/Berry/Jahn-Teller/Renner-Teller effects have their origin in degenerate (molecular) states which form singular NACTs that are poles (Section 5.1 in the BOOK). The unique feature characteristic of these molecular systems is the fact that the poles are not isolated points but arrange themselves along infinite long lines in configuration space (Chem. Phys. Lett. 322, 520 (2000); Ref. 245; Chem. Phys. Lett. 349, 84 (2001); Ref. 255; J. Chem. Phys. 115, 3673 (2001); Ref. 258; J. Phys. B 37, 4603 (2004); Ref. 297 J. Chem. Phys. 125, 094102-1 (2006); Ref. 307).
(2) We proved that the NACTs behave as fields inside the molecules. These are quantum-mechanical fields (and they are, therefore, expected to be weak), governed by an extended version of the Maxwell equations, namely, the non-Abelian Maxwell equations. The first equation is the Curl equation given in Eq. (4) and can be written as follows:
The second equation, just like the Curl equation, has its origin in the ADT (Chem. Phys. Lett. 35,112 (1975); Ref. 34) and takes the form
We showed that these Maxwell equations (once solved) yield fields that are identical to the NACTs formed by ab initio calculations (J. Chem. Phys., 121, 4000 (2004); Ref. 291; Int. J. Quant. Chem., 99, 594.(2004); Ref. 290).
(3) One of the more interesting magnitudes that emerge from the theory is the topological spin. We showed that these spins have the features that characterize the ordinary electronic spins (Chem. Phys. Lett. 329, 450 (2000); Ref. 243).
(4) In general the ADT leads to a substantial number of diabatic nuclear equations that have to be solved. Frequently most of the corresponding a priory adiabatic states are energetically closed and therefore it is enough to consider only a small number of states. Indeed, a study for this purpose led to a formalism that yields, the more relevant two diabatic states (J. Phys. Chem. A 109, 3476 (2005); Ref. 299). Since any two-state ADT is characterized by the ADT angle, g12 (see Eq. (4)) it can be shown that this angle is given either as:
where A11 (A22) and A21 (A12) are the corresponding elements of the N´N ADT matrix given in Eq. (3). It can be seen that in case of N=2 we get that = and that the result is exact.
All the findings that follow from the theoretical considerations and are supported by extensive model studies (see Refs. 215, 218, 239, 243, 245, 248 and also Chapter 3 of the BOOK mentioned below) and numerous ab-initio calculations (employing MOLPRO) carried out for tri-atomic systems, i.e., H+H2 (Refs. 268, 278, 292); C2H (Refs. 249, 258,259, 260, 272, 279); NH2 (Ref. 301); H2O (Refs. 296, 298); NaH2 (Ref. 282); NHN (Refs. 302, 305) and tetra-atomic systems, e.g, C2H2 (Ref. 300); C2H2+ (Ref. 306). Some of these systems are discussed in the BOOK (mainly in Chapter 4)
The ideas that were briefly summarized in the previous pages are extensively discussed in my Book: "Beyond Born Oppenheimer: Electronic Non-Adiabatic Coupling Terms and Conical Intersections" published by Wiley on April 21, 2006.
II. Quantum Mechanical Treatment of Reactive Processes
II.1 Negative Imaginary Potentials and Absorbing Boundary Conditions
In 1988 I considered, together with Dr. D. Neuhauser, (who in those days made his first steps in molecular physics) the possibility that Negative Imaginary Potentials (NIPs) may serve as an absorber of molecular flux. To be of any use such a NIP has to inhibit (almost completely) the passage of an outgoing molecular flux and, simultaneously, to reflect only a negligible amount of it. This study was carried out for a linear NIP of the type:
(thus it is defined in terms of two parameters) and we obtained analytic expressions to guarantee that the above-mentioned features for a system with a given mass m and a (kinetic) energy E are indeed fulfilled:
This study is reported in: J. Chem. Phys., 90, 4351 (1989); Ref. 148): This is not the first article in molecular literature that mentions NIPs but it is the first to discuss the applicability and reliability of NIPs for carrying out molecular dynamics calculations. In a series of studies we showed how to incorporate the NIPs within time-dependent wave packet frameworks (J. Chem. Phys. 91, 4651 (1989); Ref. 151) and time independent frameworks (J. Chem. Phys. 92, 3419 (1990); Ref. 160; J. Chem. Soc. Faraday Trans., 86, 1721, 1990; Ref. 161). In these studies we showed how to manipulate the absorbing boundary conditions (ABC) in order achieve arrangement decoupling in case of multi-arrangement (reactive/exchange) processes. Our publications, are the first to report on accurate reactive probabilities obtained employing only the reagents arrangement (usually one needs for this purpose also products arrangements).
Once having this method we published the first accurate three dimensional quantum mechanical integral cross sections for the most important reactive system, namely F+H2(vi=0, ji=0,1) ® HF+H (Chem. Phys. Lett. 176, 546 (1991); Ref. 172). Later we reported on the first quantum mechanical temperature dependent rate-constants for the two isotopic processes; F+X2®XF+X; X=H,D (Chem. Phys. Lett. 257, 421 (1996)); Ref. 212) and finally presented the first state-to-state differential cross sections which were found to in excellent agreement with the experimental ones as measured by the Toennies group (J. Chem. Phys. 104, 2743 (1995); Ref. 210).
In addition let me mention the quantum mechanical energy dependent integral cross sections for the ionic system Ne+H2+(vj=0,1,2) ® NeH+ + H (J. Chem. Phys. 110, 6278 (1999); Ref. 221). These results were found to be in good agreement with Ng's group measurements carried out a few years later (J. Chem. Phys. 119, 10175 (2003)).
We also carried out studies related to the tetra-atomic systems, such as H2+OH«H2O+H (J. Chem. Phys. 107, 3521 (1997); Ref. 223), O3+O®O2+O2 (J. Chem. Phys. 102, 3474 (1995); Ref. 203)) and NH(D)+OH® products (J. Phys. Chem.A. 102, 10455 (1998): Ref. 233).
The NIP-ABC approach is nowadays the most popular method to treat ordinary and complicated reactive exchange processes. However the 'user friendliness' of this approach makes the NIPs also popular in many other fields. Just to mention a few: the semi-classical and the quantum mechanical treatments of the micro-canonical density operators, molecular conductivity, electron scattering, ionization processes and even exchange processes within nuclear physics. There exists an extensive review by Muga et al. that tells all the details (Phys. Rep., 395, 357 (2004)).
II.2 The Application of Toeplitz Matrices to Scattering Problems
In 1993 I started working with Dr. M. Gilibert (a young scientist from the university of Barcelona) on a method that treats scattering processes employing features that characterize Toeplitz matrices.
A Toeplitz matrix, A is an infinite (two-dimensional) symmetric matrix that contains, along each diagonal, identical numbers. In addition the numbers along the various diagonals decrease with the distance from the principal diagonal.
Having such a matrix we are interested in a solution of the following set of homogeneous algebraic equations Ax=0 or more explicitly:
In this expression L can be made arbitrarily large and N has to fulfill N>L+1. Since the elements ANN+j are independent of N (and since A is symmetric) the above equation can be written as:
Since the solution of this equation does not depend on N a sufficient condition to obtain a robust solution is by requiring (xN+j/xN)=bj. Substituting this expression in the above equation yields the following polynomial equation for b:
This polynomial (known also a s the Toeplits polynomial) equation is almost unavoidable if one implements the perturbative approach to treat the Schrödinger equation for nuclear scattering processes. Within this approach one presents the unknown function y as where y0 is a solution of the unperturbed (elastic or inelastic) Schrödinger equation: and c solves the resulting inhomogeneous Schrödinger equation:
More details on how this (perturbative) approach and the above mentioned Toeplitz polynomial yield the S-matrix element for a simplified elastic scattering process (in this case the relevant phase shift) can be found in J. Chem. Phys. 99, 3503 (1993); Ref. 187. Details concerning more elaborate (mainly reactive) scattering processes are discussed in Phys. Rev. A 49, 4549 (1994); Ref. 196; J. Phys. A 28, L243 (1995); Ref. 203); Chem. Phys. Lett. 244, 299 (1995); Ref. 209 and J. Chem. Phys. 105, 9141 (1996); Ref. 215.
This approach was later employed to calculate the temperature-dependent integral cross sections for the reaction H+D2 ® DH+H which were found to be in good agreement with the experiments of the Wolfrum's group (J. Chem. Phys. 106, 7654 (1997); Ref. 221)