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Back | Show only these items: | On diabatization and the topological D-matrix: Theory and numerical studies of the H + H2 system and the C2H2 molecule

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj1

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj2

This derivation enables more intimate relations between the adiabatic and the diabatic frameworks as is discussed in detail in the manuscript.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj2

(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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj3

These studies are expected to yield detailed information on the possibility of diabatization of this system.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj3

(b) We report on preliminary results concerning the C₂H₂ molecule.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj4

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con1

Introduction

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac1

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac2

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac3

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac4

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac5

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac6

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac7

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac8

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac9

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac10

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot1

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac11

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot2

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac12

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj5

The theoretical outcomes are supported by numerical calculations.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj6

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj7

As for H + H₂: this system was studied with respect to the title subject by several groups^2–12 as well as by ours.^13,14

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac13

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con2

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con3

The Hilbert sub-space

In order to be able to study interactions between Hilbert spaces we introduce the the NACT, τ_jk, which relates to two states j and k and is defined as follows: τ_jk = 〈ζ_j|∇ζ_k〉; 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 |ζ_k(s_e|s)〉; k = 1,2,…N, are the eigenfunctions of a given electronic Hamiltonian H_e(s_e|s): (H_e(s_e|s) – u_k(s))|ζ_k(s_e|s)〉 = 0; k = 1,…,NHere s_e and s are the electronic and the nuclear coordinates, respectively and u_k(s) are the usual electronic eigenvalues which are recognized as the adiabatic PESs.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod1

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod2

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod3

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res1

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, τ_jj are identically zero.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod4

As a result, eqn. (3) leads to an important relation known as the parallel transport law namely: 〈ζ_k(s_e|s)|ζ_k(s_e|s + Δs)〉 = 1 + O(Δs²)Eqns.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod5

(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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod6

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod7

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod8

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod9

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: |τ_jk| ≅ O(ε) for j ≤ N; k > NHere ε is a relatively small number.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod10

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod11

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod12

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod13

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod13

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod14

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 D^4–8,14,25 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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod15

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod16

(1) The completeness condition:

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod17

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod18

(2) The curl condition:^18,19

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod19

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod19

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod20

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac14

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod21

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac15

To this claim we say the following:

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac16

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod22

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod23

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod24

The Born–Oppenheimer approach

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac17

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.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj8

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj8

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(s_e,s) = T_n(s) + H_e(s_e|s)where s_e and s have been introduced before, T_n(s) is the nuclear kinetic energy and H_e(s_e|s), the electronic Hamiltonian, which also contains the nuclear Coulomb interactions and depends parametrically on the nuclear coordinates.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod25

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod26

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod27

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 {ψ_j; 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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod28

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod29

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⁾.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod30

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod31

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod32

The Schrödinger equation for the nuclei: The diabatic representation

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod33

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod34

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod35

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod36

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod37

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod38

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod39

We elaborate more about this issue in Section IV.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac18

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?

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj9

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod40

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod41

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.

Type: Hypothesis | Advantage: None | Novelty: None | ConceptID: Hyp1

We discuss two ways to derive this transformation.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj10

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod42

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj11

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj12

We start by considering the electronic basis sets.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj13

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(s_e|s) = A(s)ζ(s_e|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 ζ_j(s_e|s)-eigenfunctions: or in matrix notation:∇ζ^T(s_e|s) – ζ^T(s_e|s)τ(s) = 0 ⇒ ∇ζ(s_e|s) + τ(s)ζ(s_e|s) = 0

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod43

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod44

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod45

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod46

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod47

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod48

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod49

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(s_e|s)-eigenfunctions that now we intend to solve.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod50

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod51

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj14

We found two necessary conditions that have to be satisfied.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs1

(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(s_e|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(s_e|s) in the second term of eqn. (27) has to be zero.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod52

100

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod53

101

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res2

102

(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(s_e|s₀) and ζ(s_e|s|s₀) have to be the same up to a phase factor, namely: ζ_j(s_e|s₀|s₀) = ζ_j(s_e|s₀)exp(iϑ_j); j = 1,…,Nwhere ϑ_j; j = 1,…,N are real phases.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod54

103

In the case where we treat only real eigenfunctions eqn. (29) becomes: ζ_j(s_e|s₀|s₀) = ±ζ_j(s_e|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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod55

104

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

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105

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).

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con4

106

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

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107

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res4

The diabatization of the adiabatic Schrödinger equation

108

The way the ADT matrix A(s) was derived in the ‘traditional’ way^18,19 is by considering eqn. (16) and demanding that A(s) eliminates the τ-matrix or in other words diabatizes the adiabatic SE.

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109

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.

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110

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:¹⁸QÃ = 0 ⇒ ∇Ã + τÃ = 0It can be shown that if eqn. (36) has a solution, then this solution, namely Ã(s) is an orthogonal matrix.

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111

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 = Ã^†uÃThe solution of eqn. (36) can be shown to be:where Ã(s₀) is a matrix that contains boundary values.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod60

112

Since Ã 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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs2

113

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res5

114

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod61

Concluding remarks

115

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)).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac19

116

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod62

117

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(s_e,s) as defined in eqn. (14) is unaffected by the ADT.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con5

118

Thus: Ψ(s_e,s) = ζ^T(s_e|s)Ψ(s) = ζ^T(s_e|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(s_e,s) is unaffected by the ADT.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con6

119

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con7

The unifications of the diabatic representations

120

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res6

121

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res7

122

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj15

123

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res8

124

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 H_e(s_e|s) operator are all identically zero.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod63

125

Next recalling eqn. (21) we mention again the following relations: (a) On the left hand side of the electronic Hamiltonian H_e we encounter the columnζ(s_e|s) for which we have shown: ζ(s_e|s) = A(s)ζ(s_e|s₀) ⇒ ζ(s_e|s₀) = A^†(s)ζ(s_e|s)(b) On the right hand side of H_e we encounter the row ζ^T(s_e|s) for which again we apply eqn. (21)): ζ^T(s_e|s) = ζ^T(s_e|s₀)A^†(s) ⇒ ζ^T(s_e|s₀) = ζ^T(s_e|s)A(s)The relevant changes yield the following diabatic potential energy element: W_jk(s|s₀) = 〈ζ_j(s_e|s₀)|H_e(s_e|s)|ζ_k(s_e|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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod64

126

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod65

127

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs3

128

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res9

129

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac20

130

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res10

131

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res11

The topological matrix D(Γ)

132

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res12

133

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod66

134

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod67

135

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod68

136

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod69

137

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod70

138

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).

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac21

139

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac22

140

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac23

141

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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs4

142

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res13

143

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac24

Numerical studies

144

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj16

145

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.

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot3

146

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac25

147

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac26

148

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

Type: Motivation | Advantage: None | Novelty: None | ConceptID: Mot4

149

In the present article we discuss a few preliminary results.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac27

150

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod71

151

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod72

152

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod73

153

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.

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac28

154

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).

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj17

155

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod74

156

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].

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod75

157

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:D_jj(q) = D_j+1,j+1(q) = cos α_j,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}.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod76

158

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod77

159

To calculate the NACTs we employed the MOLPRO program.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod78

The H + H₂ system

160

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod79

161

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod80

162

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod81

163

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod82

164

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod83

165

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod84

166

Four circular contours are considered; three of them centered at the D_3h 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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod85

167

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod86

168

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod87

169

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs5

170

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac29

171

Here we emphasize the large values that are attached to the elements three adjacent elements τ_ϕ,j,j+1; j = 1,2,4, as compared to τ_ϕ34 as well as to all the off-tridiagonal (non-adjacent) elements τ_ϕjk 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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs6

172

The only exception is τ_ϕ13 which is relatively large.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs7

173

The reason is attributed to the strongly overlapping (1,2) and (2,3) intersections which are, essentially, the ones that produce the values τ_ϕ13 (see discussion on this subject in ^{ref. 32}).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res14

174

The various diagonal elements of the D-matrix, namely D_jj(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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs8

175

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}).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res15

176

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res16

177

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac30

178

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj18

179

As for the (4,5) intersections we encountered one D_3h ci.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs9

180

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod88

181

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs10

182

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs11

183

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 C_2v intersections that exist for the situation formed by R_HH = 0.74 Å (we may have also C_2v intersections along the symmetry line which are not, necessarily, on these circles).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod89

184

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con8

185

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs12

186

The following points are to be noticed:

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac31

187

(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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res17

188

The situation changes significantly when a third state is added.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs13

189

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res18

190

(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).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs14

191

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).

Type: Result | Advantage: None | Novelty: None | ConceptID: Res19

192

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs15

193

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs16

194

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs17

195

The conclusions of this study are as follows:

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac32

196

(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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con9

197

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con10

198

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con11

199

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con12

200

(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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con13

201

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs18

202

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res20

203

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs19

204

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res21

205

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.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj19

206

The locations of the (1,2) and (4,5) intersections are obvious (both are D_3h intersections).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs20

207

Less obvious are the positions of the two (2,3) C_2v intersections.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs21

208

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 R_HH (=0.74 Å in the present case).

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs22

209

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs23

210

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs24

The C₂H₂ molecule

211

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod90

212

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod91

213

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.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod92

214

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod93

215

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod94

216

In Fig. 3 are presented several ϕ-dependent NACTs, namely, τ_ϕ,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)).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod95

217

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

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod96

218

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res22

219

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

Type: Result | Advantage: None | Novelty: None | ConceptID: Res23

220

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs25

221

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res24

222

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.

Type: Result | Advantage: None | Novelty: None | ConceptID: Res25

223

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

Type: Background | Advantage: None | Novelty: None | ConceptID: Bac33

224

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.

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs26

225

Next we refer to the energy value at the ci.

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj20

226

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°.

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod97

227

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

Type: Observation | Advantage: None | Novelty: None | ConceptID: Obs27

228

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

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229

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

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230

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con14

Conclusions

231

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj22

232

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.

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233

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.

Type: Object | Advantage: Yes | Novelty: New | ConceptID: Obj23

234

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).

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con15

235

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.

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236

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.

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237

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con18

238

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj24

239

The major numerical study is carried out for the H₃ system and therefore can be considered as an extension of our own previous studies^13,14 as well as studies by other groups.^7–12

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj25

240

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}).

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241

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).

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con20

242

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).

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con21

243

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).

Type: Model | Advantage: None | Novelty: None | ConceptID: Mod98

244

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con22

245

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

Type: Object | Advantage: None | Novelty: New | ConceptID: Obj26

246

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con23

247

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.

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con24

248

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.

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249

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.

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250

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con26

251

(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.

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252

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

Type: Conclusion | Advantage: None | Novelty: None | ConceptID: Con28