classification of topological effects in molecular systems

2 June 2000

Ž .Chemical Physics Letters 322 2000 520–526www.elsevier.nlrlocatercplett

Classification of topological effects in molecular systems

Michael Baer )

Department of Physics and Applied Mathematics, Applied Physics DiÕision, Soreq NRC, YaÕne 81800, Israel

Received 9 March 2000; in final form 17 April 2000

Abstract

This article discusses two subjects which are shown to be closely connected. The first subject concerns the existence ofpure diabatic states and the derivation of an adiabatic-to-diabatic matrix within a sub-Hilbert space. If certain requirementsare fulfilled this sub-space behaves, for all practical purposes, like a full Hilbert space. The second concerns the necessarycondition for having uniquely defined diabatic potentials. Fulfillment of this condition has led to the introduction of a matrix– the Topological matrix – which was found to contain the topological features of a given system. In the present Letter, thestudy of the topological matrix is extended to reveal new features. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction

In two recent publications, we discussed the con-Ž .ditions to be fulfilled by a non-adiabatic matrix t sŽ .in order to yield a diabatic potential matrix W s

which is uniquely defined throughout configurationŽ . w xspace CS 1,2 . It was shown that these conditions

are very close to being quantization-type conditions,to be fulfilled by the t-matrices. A diabatic potentialmatrix is obtained from an adiabatic potential matrixŽ .u s via an orthogonal transformation performed by

Ž . Ž . w xthe adiabatic-to-diabatic ADT matrix A s 3–5 :

W s sA† s u s A s 1Ž . Ž . Ž . Ž . Ž .†Ž . Ž .where A s is the complex conjugate matrix of A s

Ž .and s is a point in CS. Since u s is uniquelyŽ .defined in CS, the uniqueness of W s is solely

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Ž .determined by the ADT matrix A s . This matrix isw xgiven in the form 4,6 :

s<A s s sexp y d sPt A s 2Ž . Ž .Ž . H0 0ž /s0

where the integral in the exponential is performedalong a line G , that combines the two points s ands , and the dot, under the integral sign, stands for the0

Ž .scalar product between the vector–matrix, t s , andd s, a differential length element along G .

Ž < .To reveal the features of A s s a closed path,0

G , defined in terms of a continuous parameter l sothat the starting point s of the path is at ls0, was0

w xintroduced 1,2 . Next b was defined as the valueattained by l, once the path completes a full cycle

Žand reaches its starting point e.g., in case of a circle. Ž . Ž .bs2p . Thus, A s 'A ls0 and, following Eq.0

Ž . Ž < . Ž .2 , A s s 'A lsb . In order for the diabatic0 0

potentials be uniquely defined, it is sufficient thatŽ < . ŽA s s is uniquely defined as well, namely, A ls0. Ž .0 has to be identical to A lsb for any point s0

0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 00 00463-2

( )M. BaerrChemical Physics Letters 322 2000 520–526 521

and for any closed path G . However, we showedthat this condition is not necessary and in factŽ < .A s s does not have to be uniquely defined in CS.0

Instead, we formulated the necessary condition to befulfilled in order to guarantee the uniqueness of thediabatic potentials. For this purpose, the followingmatrix D is defined:

DsA lsb A† ls0 3Ž . Ž . Ž .Ž .which, recalling Eq. 2 , becomes

Dsexp E d sPt . 4Ž . Ž .G

This matrix, which from now on will be termed theŽ .topological matrix TM , is seen to solely depend on

the path G but not on any particular point on thispath. Then it was proved that the TM has to be

Ždiagonal In fact, recently we found that there is oneexceptional situation where this matrix is not neces-sarily diagonal but this case will be discussed else-

.where. and must have the following terms:

D G sd exp i x G 5Ž . Ž . Ž .Ž .jk jk k

namely terms of norm 1.ŽSince the TM contains all the topological or

.geometrical features of a given system, it plays afundamental role within the fields of molecular dy-namics and spectroscopy. The aim of the presentLetter is to study in greater detail this physical–mathematical entity. For this purpose, we shall usethe fact that the ADT matrix can be shown to be anorthogonal matrix and therefore, as such, can be

w xpresented in terms of a set of ADT angles 5,7,8 . InSection 2, we discuss these angles, in Section 3 weclassify the D-matrices according to certain featuresand a summary is given in Section 4.

2. Adiabatic-to-diabatic transformation angles

Since the ADT matrix is an orthogonal matrix, itselements can be expressed in terms of cosine andsine functions of certain angles. The best knownexamples are the ordinary rotation angle in two

Ž .dimensions and the three Euler rotation angles inthree dimensions. Similar angles may be defined forany dimension M.

For the past two decades, there has been on-goingdiscussion whether diabatic potentials exist other

than for the trivial case of a complete Hilbert space.If rigorous diabatic potentials did not exist, as was

w xfrequently argued by Mead and Truhlar 9,10 , thetheoretical and numerical treatments within both dy-namics and spectroscopy would not be possible. InAppendix A it is shown that if the non-adiabaticcoupling terms formed by adiabatic electronic func-

Žtions of a sub-space of dimension M in a given.Hilbert space of dimension N with electronic func-

Žtions belonging to the outer sub-space of dimension.N–M fulfill the following conditions:

² < :t s z =z (0 for i(M ; j)M 6Ž .i j i j

then that M-dimensional sub-space can be treatedentirely independently of what happens in any partwithin the outer sub-space. Here z ; ks i, j arek

Želectronic adiabatic eigen-functions certain aspectsof this subject were already treated by Pacher,

w x.Cederbaum and Koppel 6,11 . The equation thatyields the ADT matrix A is a first-order vector

w xequation of the form 3–5 :

=AqtAs0 7Ž .where = is the gradient operator. In Appendix A it isshown, employing projection operators, that if for a

Ž .given M M-N the t-matrix elements fulfill Eq.Ž . Ž .6 , then Eq. 7 can be replaced by a ‘reduced’equation of the type:

=AŽM .qt AŽM .s0 8Ž .M

where t is an M=M anti-symmetric matrix whichM

contains the non-adiabatic coupling terms of theabove-mentioned M states and AŽM . is the corre-sponding ADT matrix for these M states. Moreover,it can be shown that AŽM . has all the requiredfeatures and in particular that it is an orthogonalŽ .unitary matrix and can be presented in terms of a

Ž .line integral similar to that given in Eq. 2 . As forthe relevant topological matrix DŽM . it can be shown

Ž . Ž .to fulfill Eqs. 3 – 5 .Before continuing, we refer to two special cases

to give an indication of what we intend to achieve.First the two-state case is treated. Here AŽ2. is of theform:

cos g sin g12 12Ž2.A s 9Ž .ž /ysin g cos g12 12

( )M. BaerrChemical Physics Letters 322 2000 520–526522

w xwhere g , the ADT angle, can be shown to be 3,5 :12

sX X

g s t s Pd s . 10Ž . Ž .H12 12s0

w xRecently we proved 1,2 that in order for a 2=2diabatic potential matrix to be uniquely defined in

Ž .CS, the angle a 'g b , where b , as men-12 12

tioned earlier, is the value of l at the end point, canattain only discrete values of the form:

a sE t Pd ssnp 11Ž .12 G 12

Ž .where n is an integer. Substitution of Eq. 11 intoŽ . Ž . Ž2.Eq. 9 and recalling Eq. 3 , we get for D the

following result:nŽ2.D s y1 I 12Ž . Ž .

where I is the unit matrix. The case where n is anodd number is recognized as the Jahn–Teller casew x12,13 , and the case when n is an even number as

w xthe Renner–Teller case 14,15 . No other values areallowed for n.

The three-state case is more complicated but wasw xrecently studied to some extent 1,2,8 . Here we

mention the main points related to our subject. Thematrix AŽ3. was presented as a product of three

w xmatrices 5,7 :

AŽ3. g ,g ,g s QŽ3. g 14Ž . Ž . Ž .Ł12 23 13 i j i ji-j

Ž3.Ž .where the matrices Q g are 3=3 rotation matri-ij i jŽ .ces which contain in the diagonal, in positions ii

Ž . Ž . Ž .and jj , the function cos g , in the third diagonali jŽ . Ž . Ž .position the unity, in positions ij and ji j/ i , it

Ž . w Ž .xcontains sin g and ysin g respectively andi j i jŽ3.Ž .the rest are zeros. For instance, Q g is of the13 13

form:

cos g 0 sin g13 13Ž3.Q g s . 15Ž . Ž .0 1 013 13 � 0ysin g 0 cos g13 13

Ž .If the order of the product in Eq. 14 is taken asŽ . Ž . Ž .12 = 23 = 13 it can be shown by a direct multi-plication that the AŽ3.-matrix is:

c c ys s s s s c s qc s c12 13 12 23 13 12 23 12 13 12 23 13Ž3.A s ys c yc s s c c ys s qc s c12 13 12 23 13 12 23 12 13 12 23 13ž /yc s ys c c23 13 23 23 13

16Ž .Ž . Ž .where c scos g and s ssin g .i j i j i j i j

w x Ž .In Ref. 8 we derived, employing Eq. 8 for theassigned order of multiplication, the three differentialequations for the three ADT angles which, for thesake of completeness, are presented here as well:

Ž .=u syt y tan u yt cos u qt sin u ,12 12 23 13 12 23 12

Ž .=u sy t cos u qt sin u , .23 23 12 13 12

y1Ž . Ž .=u sy cos u yt cos u qt sin u13 23 13 12 23 12

17Ž .Ž3.Ž .Assuming that A s is chosen to be the unit0

matrix at a point s on the path G , then the value of0

AŽ3. at the end of the loop will be identical to theŽ3. Ž Ž .. Ž3.matrix D see Eq. 3 . As explained earlier D

Ž Ž3.has to be a diagonal matrix to guarantee that A.will form uniquely defined diabatic potentials . To

Ž3. Ž Ž3..obtain A at the end point of the loop 'D weŽ .have to integrate Eq. 17 from ls0 to lsb.

Defining the values of g at the end point, namelyi jŽ . Ž3.g b , as a , it can be shown that in order for Di j i j

to be diagonal all a -angles have to be multiples ofi jŽ .p or zero namely:

a sn p 18Ž .i j i j

Žwhere the n s are integers in what follows n 'i j i j. Ž . Ž3.n . Thus, recalling Eq. 16 the elements of Dji

take the form:

DŽ3.sd cos a cos a ; i/k/m 19Ž .i j i j i k im

or also:Ž .n qnŽ3. i k i mD d y1 ; i/k/m . 20Ž . Ž .i j i j

This expression can be easily extended to anydimension M if the AŽM . is written in the form:

My1 MŽM . ŽM .A s Q g . 21Ž . Ž .Ł Ł i j i j

is1 j)i

ŽM .Ž .Here the Q g are M=M dimensional matricesi j i jŽ3.Ž .which have a similar structure to Q g , namely,i j i j

in the diagonal two cosine functions and the rest ofŽ . Ž .the My2 elements are q1 s; in the off-diagonal

positions we have the two sine functions at therelevant positions and the rest are zeros. Since all

Žrelevant a -angles have to be multiples of p ori j. ŽM .zero for any M, the elements of D will be:

MŽM .D sd cos a 22Ž .Łi j i j i k

k/i


or alsoM

ni kÝk/ iŽM .D sd y1 ; is1, . . . M 23Ž . Ž .i j i j

which completes the derivation of the TM for themost general M-dimensional case. It is important to

Ž .reiterate that in Eq. 23 it is assumed that n 'n .i k k i

3. Classification of topological effects

The fact that DŽM ., a matrix which contains thetopological features of a given system, can be repre-sented in terms of certain M integers may, possibly,lead to classification of topological effects. In thissection, we make a first attempt towards such aclassification. In order to simplify the presentationwe generalize the meaning of two concepts:

Ž .a In Section 2, it was shown that a Jahn–TellerŽ .JT system is a two-state system for which the two

Ž2. Ž .diagonal elements of D are equal to y1 . In whatŽ .follows any system for which all diagonal ele-

ments of DŽM . are negative will be defined as a JTsystem.

Ž .b In Section 2, it was shown that a Renner–TellerŽ .RT system is a two-state system for which the two

Ž2. Ž .diagonal elements of D are equal to q1 . In whatŽ .follows any system for which all diagonal ele-

ments of DŽM . are positive will be defined as a RTsystem.

In the two-state case we encounter only these twoŽ Ž .possibilities the third possibility of having a y1

Ž . Ž Ž ..and q1 does not exist by definition see Eq. 12 .However, when M)2 we may encounter systemsother than just pure JT and RT systems, namely

Ž .systems where some of the diagonal elements arepositive and some are negative.

It seems to us that pure JT systems are those withthe strongest topological features, where all eigen-functions change sign upon following a closed con-tour. Therefore it would be interesting to examinethe conditions for having a JT system for an arbitrary

Ž .value of M. From Eq. 23 it is noticed that for agiven M, the condition for a JT system is that forevery 1( i(M we have:

M

n s2m q1 24Ž .Ý i k ii/k

where m is an integer. Summing both sides over ii

yields:M M M

n s2 m qM . 25Ž .Ý Ý Ýi k iis1 k/i is1

Since it was assumed that n 'n , the expressioni j jiŽ .on the left-hand side in Eq. 25 can also be written

as:M M My1 M

n s2 n 26Ž .Ý Ý Ý Ýi k i kis1 k/i is1 k)i

which implies that this expression is always even,irrespective of the value of M. Now, comparing the

Ž . Ž .right-hand-side expression in Eq. 25 with the evenŽ .right-hand-side expression in Eq. 26 , it is noticed

that these two can be equal if, and only if, M is aneven number. Therefore, pure JT systems exist if,

Ž .and only if, the dimension M of a sub-Hilbertspace is an even number. In other words odd JTsub-spaces do not exist. For instance for a two-statesystem we have, of course, the JT situation but in athree-state case no JT situation can be formed. Incontrast to the JT systems, the RT systems always

Ž .exist the proof is straightforward .As for the in-between systems in those cases too,

not all of them exist, for a given M. We alreadymentioned earlier that an in-between case does notexist for Ms2. But they may exist for other valuesof M. For instance, in the case of a three-statesystem, we may have the situation where the DŽ3.

Ž . Ž .contains two y1 s and one q1 but never theŽ . Ž .situation where it contains two q1 s and one y1 .

Ž .In short, we enconter here the situations: 3,0 andŽ .1,2 where the first digit stands for the number ofŽ . Ž .q1 s and the second for the number of y1 s.

Next we refer to the case of Ms4. In this case,the four diagonal elements of DŽ4. are given in termsof the following product of cosine functions, namely

DŽ4.sd cos a cos a cos a ; i/k/m/n .i j i j i k im in

27Ž .

Recalling that the cosine functions can be eitherŽ . Ž .q1 or y1 , it turns out that only three types of

Ž4. Ž . Ž . Ž .D matrices exist, namely, 4,0 , 2,2 and 0,4 ,where the first and the last are JT and RT matrices,respectively, and the second is an in-between case.

Ž .The two other types of matrices, namely 3,1 and


Ž .1,3 do not exist. In the same manner one will findŽ . Ž .for Ms5 the following possibilities: 5,0 , 3,2 and

Ž . Ž . Ž .1,4 and for Ms6 the possibilities: 6,0 , 4,2 ,Ž . Ž .2,4 and 0,6 .

In this respect, it is interesting to mention a recentstudy of cyclic phases of an n-fold degeneracy as

w xperformed by Manolopoulos and Child 16 . Thisstudy although restricted to n-fold degeneracy at a

Žsingle point is could be related to our subject al-though the present study refers to ADT matrices and

.ADT angles . In this respect, we found that ouranalysis and theirs yield similar results only for oddM-values. In case of even M-values, we encountermore possibilities. For instance in case of Ms2 we

Ž .have also the 2,0 situation; in case of Ms4 weŽ . Ž .have the 2,2 situation in addition to their 4,0 and

Ž .0,4 situations, and for Ms6 we have, in fact, twoŽ . Ž .additional situations, namely, 6,0 and 2,4 .

4. Conclusions

In this Letter, we discussed two subjects related toelectronic non-adiabatic coupling terms.

Ž .a The first subject is concerned with the exis-tence of pure diabatic states and a reduced ADTmatrix for a sub-space formed by negligibly smallnon-adiabatic couplings between states within thesub-space and states outside it. This subject wastreated in Appendix A where it was proved that forall practical purposes this sub-space behaves as if it

Ž .is an isolated full Hilbert space. This justifies theapplication of line integrals to construct ADT matri-ces which in turn may yield uniquely defined dia-batic potentials within this sub-space. This derivationand the subsequent findings contradict claims by

w xMead and Truhlar 9,10 that diabatic states do notexist except for the trivial case of a full Hilbertspace. In fact we are not the first to confront these

Ž w x.claims see, e.g., Ref. 11 but here, to our knowl-edge, is given the first rigorous proof for the exis-tence of an ADT matrix for a sub-Hilbert space. Inthis respect we would like to refer to a recent study

w xof the C H molecule by Mebel et al. 17 in which it2

was shown, unambigously, that in this molecule thetwo lowest states are entirely decoupled from all

Ž .other states and that all line integrals fulfill Eq. 11 .

Ž .b The second subject concerns the necessarycondition for having uniquely defined diabatic poten-tials. The requirement that the diabatic potentialsformed via the ADT matrix uniquely defined in CSled to the introduction of the topological matrix Dwhich was found to contain the topological features

w xof the system 1,2 . In this Letter we not only studiedin detail the structure and features of this matrix butalso performed the first classification of the possibletopological effects for an arbitrary system in anM-dimensional sub-Hilbert space.

Appendix A. The adiabatic-to-diabatic transfor-mation for a sub-Hilbert space

In all our treatments to-date the ADT was derivedemploying the assumption that our space is a com-

Ž .plete Hilbert space of dimension N which allowsw xthe resolution of the unity operator 2–5 . In this

appendix, the ADT will be obtained for a sub-spaceof dimension M which is not complete. The deriva-tion is done for a situation where the non-adiabaticcoupling terms that are formed by the electroniceigen-functions belonging to this sub-space withelectronic eigen-functions belonging to the outersub-space fulfill the following relations:

Ž1. ² < :t st s z =z s0 for i(M ; j)M . I.1Ž .i j i j i j

Such an assumption led Pacher et al. to introducew xtheir block diagonalization approach 6,11 . We pre-

sent our derivation in this respect as a basis forŽderiving the ADT matrix for a sub-Hilbert space a

.derivation given here for the first time .The derivation is done employing projection oper-

ators.We start by proving that if for the above two

states i and j t st Ž1.s0 the same applies to t Ž2.i j i j i j

defined as:

Ž2. ² < 2 :t s z = z s0 for i(M ; j)M . I.2Ž .i j i j

For this purpose we consider for i and j, the follow-ing derivative:

Ž1. ² < : ² < : ² < 2 :=t s= z =z s =z =z q z = z . I.3Ž .i j i j i j i j


Ž . Ž1.Due to I.1 we have =t s0 and therefore we geti j

for t Ž2. the following result:i j

Ž2. ² < :t sy Dz =z . I.4Ž .i j i j

Next we make use of the resolution of the unity andŽ .rewrite Eq. I.4 as follows:

MŽ2. ² < :² < :t s =z z z =zÝji j k k i

ks1

N

² < :² < :q =z z z =z . I.5Ž .Ý j k k iksMq1

It is noticed that, indeed, t Ž2.s0 because the firsti j

summation is zero due to the second term in eachproduct and the second summation is zero due to thefirst term in each product.

Ž .Our next aim is to derive Eq. 8 . For this pur-pose, we consider the first M coupled Born–Op-

Ž .penheimer equations which due to Eq. I.1 and Eq.Ž .I.2 can be written as:

12y = c q u s yE cŽ .Ž .j j j2m

M1Ž1. Ž2.y 2t P=c yt c s0;Ž .Ý ji i ji i2m is1

js1, . . . , M . I.6Ž .Next are defined the following projection opera-

tor, P :M

M

< :² <P s z z I.7aŽ .ÝM j jjs1

and the complementary projection operator, Q :M

Q s1yP . I.7bŽ .M M

Ž .Having these operators we can simplify Eq. I.6 byexpressing t Ž2. in terms of t Ž1.. To do that wei j i j

Ž .consider Eq. I.3 for which the first term on theright-hand side can be written as:

² < : ² < < :Dz =z s =z P qQ =zj i j M M i

which for i, j(M becomes:

² < : ² < < :=z =z s =z P =zj i j M i

M

² < :² < :s =z z z =zÝ j k k iks1

or:2Ž1.² < :=z =z s t ; i , jFM I.8Ž .Ž . jij i M

where t Ž1. is of dimension M=M. Therefore, withinM

the M th sub-space the matrix t Ž2. can be presentedM

in terms of t Ž1. in the following way:M

2Ž2. Ž1. Ž1.t s t q=t I.9Ž .Ž .M M M

Ž Ž2. Ž1.just as t is expressed in terms of t in case of a. Ž .full Hilbert space . Substituting Eq. I.9 into Eq.

Ž .I.6 yields the final form of the BO equation for asub-space of dimension M:

1 12 ŽM . 2 ŽM .y = C q u y t yE C ,M Mž /2m 2m

1ŽM .y 2t P=q=t C s0 I.10Ž . Ž .M M2m

where the dot designates the scalar product, C ŽM . isa column matrix which contains the nuclear func-

� 4tions C ; is1, . . . , M , u is a diagonal matrixi M

which contains the M adiabatic potentials and t ,MŽ1. Ž .for reasons of convenience, replaces t . Eq. I.10M

can also be written in the form:

1 2 ŽM . ŽM .y =qt C q u yE C s0Ž . Ž .M M2mI.11Ž .

which is writing the BO coupled equations in a morecompact way. A similar equation was derived by

w xPacher et al. 6,11 .Our next task is to obtain the ADT matrix AŽM ..

For this purpose we shall replace C ŽM . by F ŽM .

where the two are related as follows:

C ŽM .sAŽM .F ŽM . . I.12Ž .ŽM . Ž .Replacing C by Eq. I.12 yields the following

series of steps

2 ŽM . ŽM .=qt A FŽ .M

s =qt =qt AŽM .F ŽM .Ž . Ž .M M

s =qt AŽm.=F ŽM .q =AŽM . F ŽM .Ž . Ž .ŽM

qt AŽM .F ŽM . .M

s2 =AŽM . P=F ŽM .qAŽM .=

2 F ŽM .Ž .q =

2AŽM . F ŽM .q =t AŽM .F ŽM .Ž . Ž .M

q2t =AŽM . F ŽM .q2t AŽM .=F ŽM .Ž . Ž .M M

qt 2 AŽM .F ŽM .M


which can be further evaluated to become:

[s=2 F ŽM .q2 =AŽM .qt AŽM . P=F ŽM .Ž .M

q t q= P =AŽM .qt AŽM . F ŽM .Ž .� 4Ž .M M

where the =, in the third term, does not act beyond�4 ŽM .the curled parentheses . If, now, A is chosen to

be a solution of the following equation

=AŽM .qt AŽM .s0 I.13Ž .M

Ž .then the above kinetic energy expression becomes:

[sAŽM .=

2 F ŽM .

Ž .so that Eq. I.11 becomes:

1ŽM . 2 ŽM . ŽM . ŽM .y A = F q u yE A F s0Ž .M2m

or assuming that the AŽM . is a regular matrix wefinally obtain the Schroedinger equation within thediabatic representation:

12 ŽM . ŽM .y = F q W yE F s0 I.14Ž . Ž .M2m

Here, W defined asM

y1ŽM . ŽM .W s A u A I.15Ž . Ž .M M

is the diabatic potential matrix which, in contrast tou , is a full matrix. Since t is an antisymmetricM M

matrix for any M the matrix AŽM . can be shown tobe a unitary matrix and therefore W can also beM

written as:

W sAŽM .†u AŽM . I.16Ž .M M

ŽM .† Ž .where A is the complex conjugate matrix ofAŽM ..

Summary: In this appendix we showed that if thet-matrix elements fulfill, for a given M, the condi-

Ž .tions as specified in Eq. I.1 the M-sub-space be-haves for all practical purposes as if it is a fullHilbert space.

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classification of topological effects in molecular systems

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