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.


The theory of electronic non-adiabatic effects is usually considered as part of the broader field related to the dynamics of molecular systems.

This is indeed the case as long as one assumes that the nuclear motion is governed by model potentials or semi-empirical potentials.

However once the molecular dynamics is to be treated by ab-initio magnitudes such as potential energy surfaces (PES) and non-adiabatic coupling terms (NACT), this field extends into quantum chemistry.

As a result it becomes an interdisciplinary field located in between quantum chemistry and molecular dynamics.

The main reason for this observation is the fact that the Born–Oppenheimer eigenfunctions1 depend on two sets of coordinates the electronic coordinates and the nuclear coordinates.

The quantum chemistry treatment is carried out with respect to the (fast) moving electrons but for fixed nuclei and therefore the spatial dependence, or configuration space, is rarely considered.

On the other hand molecular dynamics is carried out with ‘average’ magnitudes, indeed derived from ab-initio treatments, but for which the electronic coordinates are not apparent either due to the way the calculations are made, as in the case of PESs, or are eliminated by integration, as in case of NACTs and also other magnitudes e.g. transition dipole moments.

One of the more important mathematical concepts that play a role in quantum chemistry is the fact that the electronic eigenfunctions form a Hilbert space.

Since while treating molecular dynamics we do not encounter, explicitly, the electronic eigenfunctions, the Hilbert space, so it seems, plays a secondary role.

This is indeed the case as long as the only interest is in PESs and similar magnitudes such as transition dipole moments etc. The situation changes significantly when NACTs are considered, for the following reason: as long as we are interested in PESs and the like we refer to magnitudes that are formed within one single Hilbert space, namely a Hilbert space at a given point in configuration space.

However NACTs, as we show next, are related to spatial derivatives (namely derivatives with respect to nuclear coordinates) and therefore are associated with nearby Hilbert spaces.

This forces us to relate to interactions between Hilbert spaces and this is the main reason for making the study of electronic non-adiabatic effects more complicated than, for instance, single-surface studies that avoid them.

Still, as long as the complete Hilbert space is small, containing a few eigenfunctions, this situation may cause some inconvenience but usually can be treated first mathematically and then numerically.

In case the Hilbert space is very large or even infinite this situation may cause insurmountable complications unless the Hilbert space breaks up into several subspaces, each comprising a small group of states.

This breakup is not expected to happen in the whole configuration space but may happen in a given region in configuration space.

The definition of a subspace of this kind, the possibility of forming Hilbert subspaces and the methods to determine the size of a Hilbert subspace in a given region is the subject of the present article.

The theoretical outcomes are supported by numerical calculations.

For this purpose we present results as obtained for the H + H2 molecular system and the C2H2 molecule.

As for H + H2: this system was studied with respect to the title subject by several groups2–12 as well as by ours.13,14

Here we present results due to additional new calculations which, further, support our theoretical findings in general and strengthen our conclusions concerning this system, in particular.

As for C2H2, this is our first attempt to study a four-atom system, it will be shown that, also in this case, the results support the theoretical findings.

The Hilbert sub-space

In order to be able to study interactions between Hilbert spaces we introduce the the NACT, τjk, which relates to two states j and k and is defined as follows: τjk = 〈ζj|∇ζk〉; k, j = {1,2,…,N}where ∇ is the grad operator (with respect to the nuclear coordinates), N stands for the number of the functions in the group and |ζk(se|s)〉; k = 1,2,…N, are the eigenfunctions of a given electronic Hamiltonian He(se|s): (He(se|s) – uk(s))|ζk(se|s)〉 = 0; k = 1,…,NHere se and s are the electronic and the nuclear coordinates, respectively and uk(s) are the usual electronic eigenvalues which are recognized as the adiabatic PESs.

To form the connection between two nearby Hilbert spaces we consider the function |ζk(se|s)〉 at the point s + Δs:15 where δkj is the Kronecker delta function.

Eqn. (3) is always fulfilled if the Hilbert spaces at a point s and the nearby region are N-dimensional (namely, contain N eigenfunctions).

However we show later that this relation holds, under certain conditions, also for smaller groups of states, namely, Hilbert subspaces.

In what follows we consider only real eigenfunctions and for them it is easy to show that the diagonal elements of the τ-matrix, namely, τjj are identically zero.

As a result, eqn. (3) leads to an important relation known as the parallel transport law namely: 〈ζk(se|s)|ζk(se|s + Δs)〉 = 1 + Os2)Eqns.

(3) and (4) may have some implications for a complete Hilbert space but they are tremendously important for situations where a smaller group of states, later termed as the Hilbert subspace, fulfils these requirements.

The importance of eqn. (3) is that the connection between a Hilbert space at a given point s and a Hilbert space in its close proximity are formed by the NACTs.

Therefore if we are interested in having a situation where eqn. (3) is fulfilled not only for a complete Hilbert space but also for a smaller group of states the way to achieve it is to demand that the τ-matrix breaks up into blocks of the same size at the point s as well as at its surroundings.

In what follows we consider a group of N states (out of an infinite Hilbert space) and for the sake of convenience we assume them to be the N lowest states, a limitation that can be easily removed.

The breakup of the Hilbert space is based on the features that characterize the above mentioned N states and are related to the NACTs, namely: |τjk| ≅ O(ε) for j ≤ N; k > NHere ε is a relatively small number.

In other words the NACTs between states that belong to the group and those outside the group are all assumed to be negligibly small.

This implies that the τ-matrix has the following form:

If this breakup takes place at every point in a given region we define the N states as a Hilbert subspace in this region.

The main difficulty with this definition is that no absolute sensible measure is given for the allowed magnitudes of the various ε's.

All we can say is “Here, ε is a relatively small number” but this statement has no meaning in practical applications.

Nevertheless the situation described by eqn. (5) is the basic assumption of the theory (and the basic interpretation for the numerical results to follow) presented in this article as well as in numerous previous publications.

In numerical applications, the measure whether a group of states forms a Hilbert subspace or not, is never deduced from eqns. (5) and (6) but is determined by much more sensitive mathematical tools, namely, the topological matrix D4–8,14,25 which is derived (in a new way) in Section III.3.1 and discussed briefly in Section V and sporadically by examining the curl condition.4b,25a

It is important to emphasize that assuming the validity of the breakup of the τ-matrix guarantees the existence of the following two conditions for a Hilbert subspace, namely the completeness condition and the curl condition.

(1) The completeness condition:

The completeness condition is an algebraic relation between the second-order NACT τ(2)jk defined as: τ(2)jk = 〈ζj|∇2ζk〉; k, j = {1,2,…,N} and the previously introduced (‘first-order’) NACT τjk which in matrix notation is written in the form:18,19τ(2) = τ2 + ∇τ

(2) The curl condition:18,19

Considering two (nuclear) Cartesian coordinates p and q we introduce the following tensorial vector Fpq: where τx, the x-component of τ, is defined as (see eqn. (1)): and [τp,τq] is the commutation relation between τp and τq.

It was proved that in case the group of states forms a Hilbert subspace, F has to be, at least approximately, zero namely: F ≃ 0The curl condition as presented here is of major importance in general physics as discussed elsewhere.20

These two conditions are not proved here because proofs were given elsewhere.

As will be shown later the curl condition, i.e. eqn. (11) is essential for any further development of the theory and therefore can be considered as the more fundamental equation in the theory of molecular non-adiabatic effects.

The assumption made in eqn. (5) was criticized several times (by the same authors) orally and in the open literature (see ref. 16) as being non-physical because it leads to strictly diabatic states which according to the critics do not exist.17

To this claim we say the following:

(1) This assumption does not necessarily lead to strictly diabatic states but it leads to approximately diabatic states.

(2) It is important to emphasize that strictly diabatic states do not exist for molecular systems, which are based on ab initio treatments.

Moreover we do not need them for any practical purposes because all final results, whether being scattering cross sections or spectroscopic cross sections are based on convergence test, calculations which can be, reliably, carried out employing approximate diabatic states.

The Born–Oppenheimer approach

Parts of the material in this chapter have been published before and therefore may be considered as redundant.

However, we intend to present new derivations for some of the previous results: derivations which are closer to quantum chemistry and therefore can be considered as more fundamental and eventually also more comprehensible.

Thus, for the sake of completeness, we also present, as briefly as possible, some parts of the theory that have been published before.

The Schrödinger equation for the nuclei: The adiabatic representation

The Hamiltonian, H, of the nuclei and the electrons is usually written in the following form: H(se,s) = Tn(s) + He(se|s)where se and s have been introduced before, Tn(s) is the nuclear kinetic energy and He(se|s), the electronic Hamiltonian, which also contains the nuclear Coulomb interactions and depends parametrically on the nuclear coordinates.

The Schrödinger equation (SE) to be considered is of the form: (H – E)Ψ(se,s) = 0where E is the total energy and Ψ(se,s) is the complete wave function which describes the motion of both the electrons and the nuclei.

Next we employ the Born–Oppenheimer–Huang expansion: where the ψj(s), j = 1,…,N are nuclear-coordinate-dependent coefficients (recognized later as the nuclear wave functions) and ζj(se|s), j = 1,…,N are the electronic adiabatic eigenfunctions of the above introduced electronic Hamiltonian (see eqn. (2)).

Substituting eqns. (12), (14) and (2) in eqn. (13) and employing the usual algebraic procedures yield the following SE for the nuclei:18,19 where Ψ(s) is a column vector that contains nuclear functions {ψj; j = 1,…,N} introduced in eqn. (13), u is a diagonal matrix which contains the adiabatic potentials, τ(2) is the second order NACT introduced in eqn. (7) and the dot designates the scalar product.

It is important to emphasize that eqn. (15) is valid for any group of states.

However if the group of states forms a Hilbert subspace then, and only then, it takes its beautiful simple form:19,21 To derive eqn. (16) we employ the completeness condition (see eqn. (8)) to eliminate τ(2).

It is important to realize that for molecular systems that contain singular NACTs eqn. (15) cannot be solved and therefore is of no practical use.

Eqn. (16) is known as the nuclear SE within the adiabatic framework and therefore is termed the adiabatic equation for the nuclei (in contrast to the diabatic equation to be introduced next).

The Schrödinger equation for the nuclei: The diabatic representation

Our starting equation is eqn. (14) with one difference, namely, we replace ζi(se|s) by ζj(se|s0); j = 1,… where s0 is a set of nuclear coordinates for a fixed point in the region of interest.

Thus, instead of the expansion in eqn. (14) the function Ψ(se,s) is presented in a slightly different form: Here j(s|s0), the corresponding nuclear coefficient, depends parametrically on s0 and ζj(se|s0), just like ζj(se|s), is an eigenfunction of a similar Hamiltonian (He(se|s0) – uj(s0))ζj(se|s0) = 0where uj(s0), j = 1,…,L are the corresponding electronic eigenvalues as calculated for this (fixed) set of nuclear coordinates.

Substituting eqns. (12) and (17) in eqn. (13), recalling eqn. (18) and continue employing the usual algebraic procedures, yield the following SE for the nuclei: where we used the fact that s0 is a constant (and not a variable).

Here V(s|s0) is the diabatic potential matrix (which, in contrast to u(s) in eqn. (16), is a full matrix) with the element Vjk(s|s0) given in the form:22Vjk(s|s0) = 〈ζj(se|s0)|He(se|s)|ζk(se|s0)〉; j,k = 1,…,Land (s|s0) is a column vector that contains the nuclear functions j(s|s0); j = 1,…,L.

Eqn. (19) is known as the nuclear SE within the diabatic framework and therefore is termed as the diabatic SE for the nuclei.

A comment: It is noticed that that the diabatic approach is based on the choice of a point s0 and the conclusion could be that there are ‘successful’ choices of s0 and ‘less successful’ choices of s0.

According to the approach presented in this article there is no such a preferred point in the region for which the Hilbert subspace is defined: all points are equally relevant for the diabatic presentation.

We elaborate more about this issue in Section IV.

The adiabatic-to-diabatic transformation

Having the adiabatic and the diabatic frameworks with the two different SEs the obvious question that arises is under what conditions will the two equations yield the same results?

It is noticed that in deriving the adiabatic equation (i.e. eqn. (16)) we assumed the electronic manifold to be of N functions but in deriving the diabatic equation (i.e. eqn. (19)) we assumed the electronic manifold to be of L functions.

In order to make a connection we assume that L ≡ N.

Next, it is expected that two frameworks are related via an orthogonal transformation (in particular that both contain the same number of basis functions) and this possibility will be studied next.

We discuss two ways to derive this transformation.

One is by considering the relevant electronic basis sets, namely, ζ(se|s) and ζ(se|s0) and the other by considering the two nuclear functions Ψ(s) and (s|s0), in eqns. (14) and (17), respectively.

The derivation via the electronic basis sets, essentially in the spirit of quantum chemistry, is done here for the first time and therefore will be somewhat more detailed.

The second derivation is reminiscent of previous studies related to this subject and therefore is discussed only briefly.

We start by considering the electronic basis sets.

The adiabatic-to-diabatic transformation matrix A(s|s0)

The connection between the two electronic basis sets is made via a (square) matrix A(s), as follows: ζ(se|s) = A(s)ζ(se|s0)To continue we refer the reader to eqn. (3), which is valid if and only if the group of states forms a Hilbert subspace in the region of interest, and convert it into a system of first order differential equations for the ζj(se|s)-eigenfunctions: or in matrix notation:∇ζT(se|s) – ζT(se|s)τ(s) = 0 ⇒ ∇ζ(se|s) + τ(s)ζ(se|s) = 0

In order to calculate the value of ζ(se|s) at a given point s for an initial value of ζ(se|s = s0) we have to assume a contour Γ that connects s and s0 and solve eqn. (23) along this contour.

The solution is given in the form:23where the integration is performed along Γ and we replaced ζ(se|s) by ζ(se|s|s0) to emphasize that the calculations started at s = s0.

This integration has to be carried out in a given order and therefore the symbol  is added to emphasize this fact.  is defined as the path ordering operator.

Comparing eqns. (24) and (21) we get for the A-matrix the expression:The matrix A(s|s0|Γ) is called the adiabatic-to-diabatic transformation matrix and is known by its acronym: the ADT matrix.18,19,23

Although eqn. (25) seems to be straightforward and simple, in fact it may contain inherent complications since the exponentiated line integral is not guaranteed to yield a single-valued A-matrix for an arbitrary τ-matrix.

Certainly, one may wonder whether the A-matrix has to be single-valued.

However the same exponentiated line integral may be applied to solve also eqn. (24) and here we may encounter serious difficulties because the τ-matrix elements are formed by the (single-valued) ζ(se|s)-eigenfunctions that now we intend to solve.

Therefore, in order for the theory to be self-consistent the τ-matrix has to produce, up to a sign, the original ζ(se|s)-eigenfunctions.

Thus, our next step is to see under what conditions this will happen.

We found two necessary conditions that have to be satisfied.

(a) Recalling eqn. (23), we consider two arbitrary (Cartesian) components of this vector equation: Differentiating eqn. (26a) with respect to q, eqn. (26b) with respect to p and subtracting the first expressions from the second we get (following some algebra): Requiring now that the ‘newly formed’ functions ζ(se|s) in eqn. (24) should be analytic (which, in addition to continuity and differentiability, requires that the order of differentiation does not affect the results) implies that the expression in front of ζ(se|s) in the second term of eqn. (27) has to be zero.

This expression, recognized as the (p,q) component of F, i.e., Fpq (see eqns. (9)–(11)), is zero if and only if the group of states under consideration forms a Hilbert subspace.

Indeed the present group forms such a subspace as was assumed to begin with.

(b) Returning to eqn. (24) and applying it for a closed contour Γ we obtain: As already mentioned earlier, in order for the theory to be self-consistent, the electronic eigenfunctions ζ(se|s0) and ζ(se|s|s0) have to be the same up to a phase factor, namely: ζj(se|s0|s0) = ζj(se|s0)exp(iϑj); j = 1,…,Nwhere ϑj; j = 1,…,N are real phases.

In the case where we treat only real eigenfunctions eqn. (29) becomes: ζj(se|s0|s0) = ±ζj(se|s0); j = 1,…,NReturning to eqn. (25) and closing the contour, as just mentioned, this fact leads to a new matrix, D(Γ), namely: which, as is noted, is identical to the A-matrix calculated for the closed contour Γ.

Eqns. (28) and (29) imply that the D-matrix elements have to be of the form: D(Γ)jk = δjkexp(iϑj(Γ)); j,k = 1,…,Nwhere we recall that the ϑj(Γ); j = 1,…,N have real values.

Summary: We have shown that the adiabatic and the diabatic electronic basis sets are related by an orthogonal transformation which is given in terms of an A-matrix presented explicitly in eqn. (25).

To satisfy self-consistency it has to have features as presented in eqns. (31) and (32) for any contour Γ in the region.

We also emphasize again that the existence of such an A-matrix with these features can be established if and only if the electronic basis set forms a Hilbert subspace.

The diabatization of the adiabatic Schrödinger equation

The way the ADT matrix A(s) was derived in the ‘traditional’ way18,19 is by considering eqn. (16) and demanding that A(s) eliminates the τ-matrix or in other words diabatizes the adiabatic SE.

In order to avoid confusion we designate the transformation matrix to be derived in this way as Ã(s) and it is assumed that Ψ(s), can be written as: Ψ(s) = Ã(s)Φ(s)where both Ã(s) and Φ(s) are yet to be determined.

Substituting eqn. (33) in eqn. (16) yields, following some algebraic arrangements, the expression:18,19 where Q is an operator defined as: Q = ∇ + τNext we require that Ã(s) fulfills the following first order differential equation:18 = 0 ⇒ ∇Ã + τÃ = 0It can be shown that if eqn. (36) has a solution, then this solution, namely Ã(s) is an orthogonal matrix.

Assuming that this is, indeed, the case then eqn. (31) becomes (following a multiplication by the complex conjugate of Ã, namely, Ã): where W(s) is the corresponding diabatic potential matrix:W = ÃThe solution of eqn. (36) can be shown to be:where Ã(s0) is a matrix that contains boundary values.

Since à is identical to A given in eqn. (25), up to up to a constant orthogonal transformation, the condition for it to be an analytic function (or matrix of analytic functions) is the fulfilment of the curl condition as given in eqns. (9)–(11).

In other words diabatization can be achieved if and only if the group of states that forms the adiabatic framework is a Hilbert sub-space.

Since the two matrices à and A are identical we label, from now on, the ADT matrix as: A(s).

Concluding remarks

As mentioned earlier, in all our previous studies the ADT matrix A(s) was derived with the aim of eliminating the τ-matrix from the adiabatic SE in eqn. (20) (see previous Section, namely, III.3.2)).

Here it was derived, independently, as a transformation matrix between two electronic basis sets: the adiabatic and the diabatic ones.

The fact that we were able to show that the corresponding electronic basis set transforms in the same way as the nuclear functions, furnishes a proof that the total function Ψ(se,s) as defined in eqn. (14) is unaffected by the ADT.

Thus: Ψ(se,s) = ζT(se|s)Ψ(s) = ζT(se|s0)Φ(s)In all previous publications (including ours) the identity between the two transformations was based on the assumption that the total wave function Ψ(se,s) is unaffected by the ADT.

In this article it is proved, for the first time, that the total wave function Ψ(se,s) is unaffected by the ADT.

The unifications of the diabatic representations

In Section II.2 we derived the diabatic potential matrix V(s) (see eqn. (20)).

In Section III.2 we derived the potential matrix W(s) which results from the ADT (see eqn. (38)).

As the last derivation we show that the two expressions yield the same result if L, the size of the diabatic group is equal to N the size of the adiabatic group which forms the Hilbert sub-space.

In other words if the Hilbert sub-space contains N functions the number of diabatic states has to be N as well.

Our starting point is the diabatic potential matrix in eqn. (38) which is written in terms of matrix elements: or recalling that eqn. (2) can also be written as:where the double sum in the second row is allowed because the off-diagonal elements formed by the He(se|s) operator are all identically zero.

Next recalling eqn. (21) we mention again the following relations: (a) On the left hand side of the electronic Hamiltonian He we encounter the columnζ(se|s) for which we have shown: ζ(se|s) = A(s)ζ(se|s0) ⇒ ζ(se|s0) = A(s)ζ(se|s)(b) On the right hand side of He we encounter the row ζT(se|s) for which again we apply eqn. (21)): ζT(se|s) = ζT(se|s0)A(s) ⇒ ζT(se|s0) = ζT(se|s)A(s)The relevant changes yield the following diabatic potential energy element: Wjk(s|s0) = 〈ζj(se|s0)|He(se|s)|ζk(se|s0)〉A comparison between eqn. (20) and eqn. (44) reveals that the two matrices, namely, W which has its roots in the adiabatic framework and follows from the ADT and V which is formed, directly, in the diabatic representation are identical.

However, while deriving the V-matrix no restrictions were imposed on the group size of the eigenfunctions, the derivation of the W-matrix requires that the corresponding eigenfunctions form a Hilbert subspace, namely, has to be of a specific size.

These facts imply that the physical diabatic framework cannot contain groups of arbitrary sizes because for some of the cases we will not be able to find a relevant single-valued adiabatic potential matrix within the adiabatic framework.

In other words: to be of any physical relevance, the diabatic framework, just like the adiabatic one, has to be made up only of groups of eigenfunctions which form Hilbert sub-spaces.

At this stage we return to our comment made at the end of Section III.2 regarding the choice of s0.

Since the various diabatic frameworks based on different s0-points are connected with one single adiabatic framework this situation rules out the existence of any preferred choice of a diabatic framework.

Practically there might exist preferred choices but these are beyond the scope of the present composition.

The topological matrix D(Γ)

As a by-product of the derivation in Section III.3.1 we obtained the D-matrix which is the title subject of the present article.

The feature of the D-matrix elements are listed in eqn. (32), namely a diagonal matrix with values for the norm equal to 1.

In case of real functions the diagonal elements of D have to be ±1.

In what follows the D-matrix is called the topological matrix.

Its existence requires that the τ-matrix satisfy the exponentiated quantization.

A matrix with this feature emerges naturally from the first derivation which is based on transforming the electronic eigenstates.

It was not realized that such a matrix should emerge from ADT treatment although the derivation was done almost three decades ago (given, briefly, in Section III.3.2).

It was only very recently revealed that by requiring that the potential matrix, W, as presented in eqn. (38) is single-valued that the τ-matrix has to be quantized.24

Since a D-matrix can be formed only by eigenfunctions of a Hilbert subspace we proposed, some time ago, to use the D-matrix as a numerical tool to find out whether a given group of states, in a spatial region which is formed by the closed contour along which D is calculated, forms a Hilbert sub-space.14

It seems to us that for a given spatial region, to calculate along its borders the relevant D-matrix is much simpler than to examine, at every point in that region, to what extent τ-matrix breaks up into sub-matrices or to check, at every point in that region, the curl condition (to examine to what extent the elements of the F-matrix become zero).

As will be shown next, large regions of configuration space can be scanned with the help of the D-matrix by doing relatively simple calculations.

It may be interesting for the reader to know that we recently completed a detailed study on this subject, namely, on the relation between the size of the Hilbert sub-space and the size of spatial region for which it applies, employing a model based on the eigenfunctions of the Mathieu equation.25a

Numerical studies

In the numerical part we concentrate on revealing the position of conical intersections (ci)26,27 and on the spatial distributions of the related NACTs of two molecular systems.

The first is the H + H2 system which is the simplest reactive system and therefore is important for the study of fundamental features of the reactive process.

In fact we have already published one detailed study on this system but it was limited to the three lowest states.14

Here the H + H2 study is extended to five states for reasons to be specified next.

The second system is the C2H2 molecule, acetylene, which is not only one of the more important molecules in chemistry but, due to recent interesting spectroscopic measurements,28 has posed a serious challenge for the theory.29

In the present article we discuss a few preliminary results.

Our general approach to reveal conical intersections for a system of M atoms is to break it up into two groups: one made up of L atoms and the other of K (= M – L) atoms where the position of the atoms in each group are frozen but the groups are allowed to move freely, the one with respect to the other.

In the present case one group is always made up of one atom and the other contains the rest (two in case of H + H2 and three in case of C2H2).

In what follows we define a situation as a combination of frozen atoms.

Although we described on various occasions the kind of calculations are done and how they are processed we repeat it here for the sake of completeness.

The aim is to derive numerically the values of the A-matrix given in eqn. (25) and the diagonal elements of the D-matrix given in eqn. (31).

As is noticed these calculations have to be carried out along given contours and like in all our previous articles the contours were assumed to be circles with a given center and a radius q.

As a result the only NACTs that are employed are the angular ones, namely: where we have dropped the (1/q) term and the only variable of integration is ϕ defined in the range [0,2π].

In the case of two states (i.e.N = 2) we are interested in the topological phase α(ϕ|q):which is connected to the 2 × 2 D-matrix as follows:Djj(q) = Dj+1,j+1(q) = cos αj,j+1(q)In the N > 2 case the D matrix is calculated by an expression similar to eqn. (31), namely, Details of how to calculate the D-matrix are given, for instance, in ref. 25b.

As results we consider the diagonal elements of the N × ND-matrix which are expected to be ±1 if the N states form a Hilbert subspace (or several Hilbert subspaces).

To calculate the NACTs we employed the MOLPRO program.

The H + H2 system

The NACTs are calculated at the state-average CASSCF level using 6-311G**(3df,3pd) basis set30 extended with additional diffuse functions.

In order properly to take into account the Rydberg states we added, to the basis set, one s diffuse function and one p diffuse function in an even tempered manner,31 with the exponents of 0.0121424 and 0.046875 respectively.

We used the active space including all three electrons distributed over nine orbitals.

Usually five different electronic states (depending on the case) including the five studied states, namely, 1 2A′, 2 2A′, 3 2A′, 4 2A′ and 5 2A′ were computed by the state-average CASSCF method with equal weights.

Convergence test were carried out with respect to the number states.

We report here on results as calculated for the situation where two hydrogen atoms are at the (fixed) distance RH–H = 0.74 Å.

Four circular contours are considered; three of them centered at the D3h point and with the radii q = 0.3, 0.4, 0.5 Å and the fourth centered at 0.25 Å further away from the H–H axis (thus, at a distance of 0.89 Å from the H–H axis along the symmetry line) with a radius q = 0.65 Å.

In Fig. 1 are presented schematically the positions of the various cis, the circular contours and the ten ϕ-dependent NACTs, namely, τϕjk(ϕ|q); j,k = 1,2,3,4,5 as calculated along the various circles.

It is important to mention that the points (q,ϕ = 0) and (q,ϕ = π) are the ‘northern’ and the ‘southern’ poles, respectively, both located on the symmetry line.

The various figures and mainly Fig. 1a, d, g and j indicate that most of the ‘action’ takes place around ϕ = π the point closest to the HH axis.

The figures essentially speak for themselves, in particular because similar ones have been presented and analyzed in previous publications.

Here we emphasize the large values that are attached to the elements three adjacent elements τϕ,j,j+1; j = 1,2,4, as compared to τϕ34 as well as to all the off-tridiagonal (non-adjacent) elements τϕjk where k > j + 1 (note the different scales of the subfigures in the two lower rows as compared to the scale of the subfigures of the upper row).

The only exception is τϕ13 which is relatively large.

The reason is attributed to the strongly overlapping (1,2) and (2,3) intersections which are, essentially, the ones that produce the values τϕ13 (see discussion on this subject in ref. 32).

The various diagonal elements of the D-matrix, namely Djj(q); j = 1,…,N as calculated for different N-values (i.e. different sizes of Hilbert subspaces) and different circles (expressed in terms of q-values) are presented in Table 1.

The results in the table clearly indicate the existence of one (1,2) ci, two (2,3) conical intersections (as was explained in detail in ref. 14).

Adding the fourth and the fifth states contributed one (4,5) ci but no conical intersections between the third and the fourth states.

The conical intersections related to the three lower states were discussed in detail in our previous publication.

Here, we start by referring to the newly exposed (4,5) ci, then we refer to the missing (3,4) ci, then we continue by briefly relating to issues discussed in previous publications and finally discuss the extent the 3 × 3 τ-matrix breaks up from the 5 × 5 τ-matrix.

As for the (4,5) intersections we encountered one D3h ci.

Its existence is proved by inspecting the values of D44(q) (=D55(q)) as calculated assuming that the fourth and the fifth states form an isolated Hilbert subspace of two states (see relevant results in Table 1 along column N = 2).

For all the four different circles we obtained values very close to –1.

Results of the same quality were obtained for D44(q) (=D55(q)) along the column N = 5.

Since it was somewhat of a mystery that the H3 system does not possess any (3,4) intersections we carried out an extensive search which, in particular, focused on the two isosceles circles (the circles with q = 0.74 Å and their centers on each of the fixed H-atoms) that are expected to go through (almost) all the C2v intersections that exist for the situation formed by RHH = 0.74 Å (we may have also C2v intersections along the symmetry line which are not, necessarily, on these circles).

Since we could not find any, our conclusion is: The H3system does not have (3,4) cis in the region of interest.

In Table 1 are presented the diagonal elements of the D-matrix as obtained not only for different circles but also employing τ-matrices of different dimensions, namely, N = 2,3,4,5.

The following points are to be noticed:

(1) It is well noticed by inspecting the results of the two lower states (i.e.D11(q) (=D22(q)) along the column N = 2) that these two states are not capable of forming an isolated two-state Hilbert subspace: the values of D11(q) are far from being –1.

The situation changes significantly when a third state is added.

It is noticed that the three lower states form, approximately, a very nice Hilbert subspace, however it slowly deteriorates as the spatial region increases (due to the increased q-values) These results were discussed extensively in our previous publication and therefore will not be discussed here.

(2) It is noticed that adding a fourth state to the three-state Hilbert subspace causes the relevant diagonal D-matrix elements to distance themselves from the expected ±1 values (cf. values along the N = 3 column and along the N = 4 column).

The main reason is that in contrast to the three states lower that form a Hilbert subspace the four states do not form a Hilbert subspace and adding the fourth state only increases the background noise (formed by terms like τ24 and τ14).

(3) However extending the four states to five states improves the situation significantly.

This is well noticed by inspecting the five diagonal elements of the D-matrix as presented in the last column (i.e.N = 5).

This addition not only improved the four-state D-matrix numbers but even the three-state D-matrix numbers became much closer to ±1.

The conclusions of this study are as follows:

(1) The five states that are the subject of the present study form two well separated (approximate) Hilbert sub-spaces: the three lower states form one Hilbert subspace and the next two states form the second Hilbert subspace.

The breakup into two groups is caused by two reasons: (1) the missing (3,4) intersections, and (2) because the τ-matrix elements that connect the three-state system to the five-state system (i.e.τ24, τ25, τ14, τ15 and τ35) are all relatively weak, as is noticed from Fig. 1.

(2) The quantization of the τ-matrix as presented through the diagonal elements of the D-matrix is undoubtedly verified in this study.

We also revealed the close relationship between N, the size of the Hilbert subspace, and q the size of the configuration space a relation that was, also, just recently discussed for a model based on the eigenfunctions of the Mathieu equation.25

(3) The results indicate that adding states of a ‘nearby’ Hilbert subspace does not necessarily improve the quantization unless one adds a complete nearby Hilbert subspace.

(4) Interesting and encouraging results are obtained for the shifted circle with q = 0.65 Å.

This shift distances the center of the previous circles, from the HH axis, by 0.25 Å so that the relevant circular region could be increased significantly without getting too close to the fixed hydrogen atoms.

It is noticed that although the spatial region surrounded by this circle is almost doubled (compared to the one for q = 0.5 Å) its diagonal D-matrix elements are of the same quality.

This implies that the slow deterioration of the nice features of the D-matrix (along the first three circles) as q increases is not necessarily connected with the size of the region surrounded by the enlarged circles but, sometimes, can be attributed to the effect (or damage) of the presence of other atoms close to this region.

Before moving to the next system we would like, for the sake of completeness, to relate to the locations of the various conical intersections and the energies at these points.

The locations of the (1,2) and (4,5) intersections are obvious (both are D3h intersections).

Less obvious are the positions of the two (2,3) C2v intersections.

Each one of them is located on a circle that has its center at the position of the respective fixed hydrogen and a radius equal to RHH (=0.74 Å in the present case).

Their location is off by 37.22° from the symmetry line (see Fig. 1).

The energies at the various intersections are: at the (1,2) ci, ∼2.7 eV, at the (2,3) ci, ∼7.0 eV and at the (4,5) ci, ∼5.8 eV.

The C2H2 molecule

As in the case of the H + H2 system, here too, the NACTs are calculated at the state-average CASSCF level using 6-311G** basis set.30

We used the active space including all ten valence electrons distributed over ten orbitals (a full valence active space).

Following convergence tests we included in the calculations, in addition to the four studied states, i.e., 1 2A′, 2 2A′, 3 2A′ and 4 2A′ also another four to six electronic states.

The calculations were carried out employing the state-average CASSCF method with equal weights.

The study of the C2H2 is carried for the situation given in Fig. 2.

In Fig. 3 are presented several ϕ-dependent NACTs, namely, τϕ,j,j+1(ϕ|q); j = 1,2,3, obtained for two circles: one with radius q = 0.3 Å (Figs. 3(a) and (b)) and the other with radius q = 0.8 Å (Figs. 3(c) and (d)).

The only topological results we report are on the values of the corresponding α12(q) (see eqn. (46)).

For the first circle we got a value α12(q = 0.3 Å) = 3.142 which is very close to π and supports the assumption that in the considered circular region we have an isolated (1,2) ci which is hardly affected by intersections due to other states.

This result implies that in the relevant circular region the two lower states form a Hilbert subspace.

The situation changes significantly when q is increased to 0.8 Å.

The value of α12 is, now, α12(q = 0.8 Å) = 1.551 which indicates that in the increased circular region the (1,2) ci is damaged by intersections between higher states, namely, τ23 and τ34.

This is also clearly shown in Fig. 3(c) which presents the ϕ-dependent angular components of these two intersections and it is well observed that at some ϕ-intervals they attain large enough values to indicate the existence of (2,3) and ((3,4) intersections in this region.

The configuration we studied was recommended by Cui et al.29a (see also Wetmore and Schafer29b) although they could not show that indeed the energy difference between the two states is zero.

In this respect we would like to add that we conducted also a detailed search for additional (1,2) cis and we could not find any.

Next we refer to the energy value at the ci.

For this purpose we consider the two geometries: (a) The ground state, collinear, configuration with the inter-atomic distances (H–C),(C–C),(C–H) = (1.076,1.218,1.076) Å (b) The bent configuration at the ci point with the inter-atomic distances (H–C),(C–C),(C–H) = (1.10,1.35,2.04) Å and the corresponding angles ∠(HCC) = 109° and ∠(CCH) = 83°.

The difference between energies related to these two geometries is 5.798 eV.

As mentioned earlier we intend to carry out a much more extensive study of the ci-distribution in this system.

We do not expect to find any additional (1,2) intersections but we identified several (2,3) and (3,4) intersections although for larger radii which seem to support the path for photodissociation as proposed by Lee et al34. and verified by Cui et al.29

If this is indeed the case then maybe one could identify the NACT between S1 and S2 as the perturber for the process.


The present article centers on a matrix that we labeled the D-matrix and termed the topological matrix.

It contains a theoretical part which presents a new derivation of this matrix and its exceptional features based purely on the spatially-dependent electronic eigenfunctions.

In other words, in contrast to the previous derivation, which was based on the diabatization of the nuclear adiabatic SE,24 here it is derived by considering solely the features of the spatial dependent electronic manifold.

Having this new derivation which is not only more fundamental but also more general we were able to make the connection between arbitrary diabatic sets of states and the corresponding adiabatic ones and in this way to enforce them (i.e. the diabatic ones) to be grouped in the same way as are grouped the adiabatic states (namely, being Hilbert subspaces).

In other words, just as each adiabatic Hilbert subspace is able to create its diabatic counterpart, the same has to apply for diabatic sets, namely, they, too, have to be grouped in such a way that they will be able to form their adiabatic counterparts.

In practice that implies that if the size of a given Hilbert subspace is N this has also to be the size of its diabatic counterpart.

This conclusion contradicts the belief regarding the arbitrary size of the diabatic groups of states and the belief in the ability to have successful and less successful choices of diabatic states.

A large part of the article is devoted to the calculated D-matrix diagonal elements (expected to be ±1 in case of Hilbert subspaces).

The major numerical study is carried out for the H3 system and therefore can be considered as an extension of our own previous studies13,14 as well as studies by other groups.7–12

One of the main results in this new study is the fact that the H3 system lacks the (3,4) ci and it is this missing ci that causes the three lower states to form a Hilbert subspace (as was established already in ref. 14).

We also revealed the fact that the next two states, the fourth and the fifth, form too, a Hilbert subspace (see first column of Table 1 where are presented the D44 and D55 (≡cos(α45))-values which are exceptionally close to –1).

Although the three lower states form approximately a Hilbert subspace, we found that considering all the five lower states as one single Hilbert subspace yields for the three lower states further improved diagonal D-matrix elements (cf.N = 3 and N = 5 columns in Table 1).

Another important comment in this respect is the following: In calculating the N × ND-matrix we always include all the N(N – 1)/2 matrix elements of the τ-matrix (presented in Fig. 1).

Trying to ignore contributions due to the smaller elements (such as the off-tridiagonal elements or the off-off-tridiagonal terms) immediately damages the results.

We also presented a few preliminary results for the interesting C2H2 molecule.

Among other things we established the existence of a (1,2) ci along the dissociative transition state region (see ref. 27).

Our calculations also hint at the possibility that in the vicinity of this ci may exist (2,3) and (3,4) intersections, a fact that may help resolve several mysteries related to the photo-dissociation process of this system.

As a final comment we would like to return to the on-going criticism16 regarding the break-up of the τ-matrix (in real molecular systems) which led to the formation of the Hilbert subspace.

The present numerical study for the H + H2 system shows unambiguously that the 3 × 3 τ-matrix is practically disconnected from the rest of the τ-matrix (as is assumed in eqns. (5) and (6)): (1) The results in Fig. 1 show that all the τ-matrix elements that connect this sub-matrix to the rest of the τ-matrix are much smaller than the others.

This applies in particular to the (essentially) missing τ34-matrix element: a fact that makes the breakup unavoidable.

(2) In Table 1 are presented the diagonal elements of the 3 × 3 D-matrix and it is well noticed that all of them are close to ±1 which, according to the theory, is due to the breakup.

The results presented here are a clear indication that the above-mentioned criticism is inappropriate and therefore it should be retracted.