curl equations as substratum for the derivation of the electronic nonadiabatic coupling terms

Curl Equations as Substratum for theDerivation of the ElectronicNonadiabatic Coupling Terms

MICHAEL BAER,1,2 ALEXANDER M. MEBEL,3 GERT D. BILLING1

1Department of Chemistry, H. C. Ørsted Institute, University of Copenhagen,2100 Copenhagen Ø, Denmark2Department of Physics and Applied Mathematics, Soreq; NRC, Yavne 81800, Israel3Institute of Atomic and Molecular Sciences, Academia Sinica, P.O. Box 23-166, Taipei 10764, Taiwan

Received 10 January 2002; accepted 3 April 2002Published online 29 August 2002 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.10333

ABSTRACT: The idea to derive the nonadiabatic coupling terms by solving the Curlequations (Avery, J.; Baer, M.; Billing, G. D. Mol Phys 2002, 100, 1011) is extended to athree-state system where the first and second states form one conical intersection, i.e.,�12 and the second and the third states form another conical intersection, i.e., �23.Whereas the two-state Curl equations form a set of linear differential equations, theextension to a three-state system not only increases the number of equations but alsoleads to nonlinear terms. In the present study, we developed a perturbative scheme,which guarantees convergence if the overlap between the two interacting conicalintersections is not too strong. Among other things, we also revealed that thenonadiabatic coupling term between the first and third states, i.e., �13 (such interactionsdo not originate from conical intersection) is formed due to the interaction between �12

and �23. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem 90: 1577–1585, 2002

Key words: Curl equations; nonadiabatic coupling terms; two- and three-statesystems; conical intersections

Introduction

I n recent years, a lot of efforts were invested instudying the nature of the electronic nonadia-

batic coupling terms (NACTs) [1–11]. The NACTs

are characterized by two features: They are vectors(in contrast to potentials, which are scalars) andthey may become singular (in contrast to potentials,which do not). If arranged in matrices, they acquirea third interesting feature, namely, the matrices areantisymmetrical.

The ordinary way to get acquainted with objectslike the NACTs is to derive them from first princi-ples, via ab initio calculations, and study their spa-

Correspondence to: M. Baer; e-mail: [email protected]. Baer is Guest Professor at the Chemistry Department,

University of Copenhagen, Copenhagen, Denmark.

International Journal of Quantum Chemistry, Vol 90, 1577–1585 (2002)© 2002 Wiley Periodicals, Inc.

tial structure—somewhat reminiscent of the waypotential energy surfaces were studied. However,this line of research is not enough. The fact that theNACTs are usually singular [12, 13] and possess afew unique features, like fulfilling Curl conditions[14, 15] and being quantized along closed contours[16, 17], indicate that the origin of these magnitudesis mathematically oriented. Assuming that thisstatement is correct, it is expected that the NACTscan eventually be derived by solving differentialequations rather than obtaining them from ab initiotreatments.

Some time ago, a connection was made betweenthe two-state Curl condition, henceforth termed theCurl equation, and a singular pseudomagnetic fieldH assumed to exist at the point of the conical inter-section (CI) [18]. This term appears in the curl equa-tion as a free term—thus converting the homoge-neous vectorial differential equation into aninhomogeneous differential equation. Once the ex-plicit form for the inhomogenuity is obtained froman ab initio treatment (or perturbation theory), thisextended Curl equation may lead, for a given gauge,to a closed formula for the NACTs. Based on thisstudy, Avery et al. [19] suggested to replace the abinitio treatment to yield the NACTs by developinggauge transformations to solve the Curl equation.In the above-mentioned article, this idea was pur-sued for a two-state system and for several CIslocated at arbitrary locations.

In this article, this treatment is extended in twoways: (1) The inhomogeneous term is deleted sothat we apply a set of homogeneous Curl equationsbut solve them for given boundary conditions thatcan be obtained from ab initio calculations; (2) thetreatment of the two-state case is applied to three-state systems.

It is important to realize that the extension fromtwo states to three states is more involved then justa simple extension because the three-state differen-tial equations, in contrast, to the two-state ones arenot linear. To solve a set of nonlinear equations ismuch more complicated and therefore we intend toshow in this article how to overcome (under certainconditions) this difficulty.

Before going into the main subject of this article,we relate to the problem of finite subspaces withinthe Hilbert space and the corresponding finite Curlequations. This is an important issue when consid-ering several electronic states and is also crucial forthe present study. In the original derivation of theCurl equation [14], it was assumed that the elec-tronic manifold forms a large and even infinite

Hilbert space. The question to be asked in thisrespect is whether a reduced Curl equation existsfor a given subspace. This subject was addressedsometime ago by Mead and Truhlar [20a] whilereferring to the possibility of constructing diabaticstates and it was claimed that strictly diabatic statescannot be formed because the Curl condition can-not be fulfilled for any finite subspace except in thecase of atom–atom systems (where it is triviallyfulfilled for any number of states). However, it isknown from numerous studies whether based onperturbation theory or ab initio treatments that ifone considers regions close enough to a given CIthe size of the subspace shrinks significantly andmay be reduced to two states [1–4, 6–10, 20b]. Thisimplies that diabatic states, not necessarily strictlydiabatic states, but accurate enough can be formedin given finite regions of configuration space andthat the “reduced” Curl equation holds, approxi-mately, in these regions. This situation was recentlyaddressed in two separate articles [21], each from adifferent point of view, and they can be summa-rized as follows: A subspace is considered in whichthe states belonging to it are strongly coupled witheach other but none of them is strongly coupled tostates outside it. As a result, the following lemma isproved: If the weak interaction between any twostates—one internal and one external—is of theorder O(�), the Curl equation related to the sub-space, i.e., the reduced Curl equation, is fulfilled upto an error of order O(�2).

In what follows, we assume, based on this lemma,that under the specified conditions applying finiteCurl equations for finite regions in configurationspace is mathematically valid.

Treatment of the Two-State System ina Plane

DERIVATION OF THE EQUATION AND THEBOUNDARY CONDITIONS

In the case of a subspace of two states that fulfillthe lemma conditions, the Curl equation takes theform [14, 15]

Curl� � 0, (1)

where, as mentioned earlier, � is a 2 � 2 vectormatrix. For the sake of completeness, we also re-mind the reader what these � matrix elements are:

BAER, MEBEL, AND BILLING

1578 VOL. 90, NO. 6

�ji � ��j��i� (2)

where the �j, j � 1, 2 are the electronic Born–Op-penheimer eigenfunctions of the electronic Hamil-tonian, which depend parametrically on the nuclearcoordinates [22], and � is the (vectorial) grad oper-ator. Equation (1) is fulfilled at any point in config-uration space (CS) for which the components of �are analytic functions.

Recalling that the 2 � 2 �-matrix contains onlyone nonzero term, �, i.e.,

� � � 0 �� 0�, (3)

the Curl equation for the matrix �, as given in Eq.(1), becomes the Curl equation for the (single non-zero) matrix element �. In other words, � is anordinary vector that fulfills the Curl equation:

Curl� � 0. (4)

Equation (4) is a set of partial first-order differ-ential of equations. In general, the number of equa-tions is equal to the number of components of thecurl equations. In this article, we consider the caseof two coordinates, namely, a planar (2-D) systempresented in terms of either Cartesian coordinates(x, y) or polar coordinates (q, � ).

We start considering the Curl equation ex-pressed in terms of polar coordinates [18, 19]:

1q ��

�q ��q

�� 0 f��

�q ��q

�� 0, (5)

where ��(q, � ) and �q(q, � ) are the angular and radialcomponents, respectively, of �. The aim, at thisstage, is to develop confidence in this equation bypresenting its ability to yield insight regarding ��(q, �)and �q(q, � ). For this purpose, we consider a case ofone single CI located at the origin of the coordinatesso that each contour (usually a circle) that sur-rounds the origin surrounds this CI.

We start by integrating Eq. (5) with regard to qalong the interval [q0, q1]:

��q1, � � � �q0

q1

dq��q

�� q0, � � (6)

and continue by integrating � along the interval [0,2�]:

�0

2�

��q1, � �d� � �0

2�

d� �q0

q1

dq��q

��

� �0

2�

��q0, � �d�. (7)

We start by referring to the first term on the leftside and the term on the right side. In both cases,the integration is done along circles (namely, con-tours with fixed q values) that surround the origin.From a detailed proof presented in Appendix C ofRef. [10] (cf. Appendix D in Ref. [15]), it was shownthat the two integrals yield the same value becausethe two circles surround the same single one CIassumed to be located at the origin. Consequently,these two terms cancel each other and we get thefollowing result:

�0

2�

d� �q0

q1

dq��q

�� 0 (8)

or, more explicitly (following change of order ofintegration),

�q0

q1

dq �0

2� ��q

��d�

� �q0

q1

dq� �q�q, � � 2�� q�q, � � 0�� 0, (9)

which implies that �q(q,� ) is single valued with re-spect to � for any q value in the range [q0, q1]. Thisby itself is an interesting outcome that cannot beproven otherwise and therefore strengthen our con-fidence in the physical–mathematical contents ofthe Curl equations.

SOLUTION FOR A SINGLE CONICALINTERSECTION

As mentioned in the Introduction, our intentionin this article is to get acquainted with the three-state Curl equation. Whereas the discussion in theprevious section was for a CI located at the origin,we consider here a CI located at an arbitrary point(namely, removed from the origin).

One of the more interesting results of Eq. (5) isthat if the radial component of �, i.e., �q, is negligi-

CURL EQUATIONS AND NONADIABATIC COUPLING TERMS

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1579

bly small (which implies that its derivatives arenegligibly small as well) then ��, the angular com-ponent of �, does not depend on q (or, at most, isweakly dependent on q).

To examine whether this is a realistic situation,we consider the well-known CI of the C2H moleculeformed by the two lowest states, namely, by 12Aand the 22A states [6, 8, 23]. In Figure 1 are pre-sented ��(q, � ) functions as calculated along circleswith different q values, all centered at the CI. The CIis located on the CC axis at a distance of 1.30 Å fromthe closer carbon, where the distance between thetwo carbon atoms is assumed to be 1.35 Å. It is wellnoted that the basic shape of ��(q, � ), while chang-ing q, is “more or less” preserved, which resemblesa weak q dependence. It is important to emphasizethat for q � 1 Å and � � the circle gets close to oneof the carbons—0.3 Å—and therefore not only theassumption that this is an isolated CI is not fulfilledanymore but also the two-state assumption breaksdown. These two deviations from our basic as-sumptions are the main cause for the significantchanges in ��(q, � ) in the relevant angular intervalof � �.

Saying that, we return to our main issue, namely,deriving the mathematical expression for a CI re-moved from the origin, located at an arbitrary point(qj0, �j0) that possesses a zero radial component.

The derivation is done as follows: We obtain thesame solution of the above CI at the origin employ-ing Cartesian coordinates and then shift the solutionto the point of interest, namely, (0, 0) 3 (xj0, yj0)(�(qj0, �j0)). Once completed, the solution is trans-formed back to polar coordinates. Because this der-ivation was done before, we present it in the Ap-pendix and not here. Following this procedure,��(q, � ) and �q(q, � ) (which is now different fromzero) become [19]

�q�q, � � � �fj��j�1qj

sin�� j�

��q, � � � fj��j�qqj

cos�� j�, (10)

where fj(�j) is a function, still to be determined, thatrepresents the angular distribution of ��(q, � ) withrespect to an origin located at the jth CI and qj and�j are the coordinates of an arbitrary point, P(q, � ),with respect to the CI position and are related to (q,� )—the coordinates with respect to the origin—asfollows:

qj � ��q cos � � qj0cos �j0�2 �q sin � � qj0sin �j0�

2

cos �j �q cos � � qj0cos �j0

qj. (11)

It was shown in a previous publication [18] [andcan be seen also from the second equation in Eq.(10)] that fj(�j) is defined as

fj��j� � �j��j� (12)

where �j0(�j) is calculated for qj0 � 0 and as such canbe obtained from ab initio calculation. We attachedto each CI a different f(� ) function, i.e., fj(�j) toindicate that each such a CI (in this case the jth one)may form a different angular dependence for itsspatial distribution.

It is noticed that for qj0 � 0 (i.e., when �j becomes�) the solution in Eq. (10) becomes �q(q, � ) � 0 and��(q, � ) � f(� ), which the solution for the case of theCI is at the origin.

FIGURE 1. The t12�(q, � )—the angular nonadiabaticcoupling term as a function of �—as calculated for theC2H molecule, for different q values. (a) q � 0.0.5 Å;(b) q � 0.2 Å; (c) q � 0.5 Å; (d) q � 1.0 Å. The cal-culations were done for a fixed CC distance, i.e., rCC �1.35 Å (see Fig. 3).


1580 VOL. 90, NO. 6

SOLUTION FOR SEVERAL CONICALINTERSECTIONS

With this modified expression, we can now ex-tend the solution of Eq. (10) to any number of CIs.The solution in Eq. (10) stands for a single CI lo-cated at an arbitrary point (qj0, �j0). Because ��(q, � )and �q(q, � ) are scalars, the solution in the case of NCIs located at the points (qj0, �j0), j � 1, . . . , N, canbe obtained by simply summing up the contribu-tions of all CIs [19]:

�q�q, � � � � �j�1

N

fj��j�1qj

sin�� j�

��q, � � � q �j�1

N

fj��j�1qj

cos�� j�. (13)

Equation (13) yields the two components of�(q, � ), the vectorial NACT, for a distribution oftwo-state CIs expressed in terms of the angular com-ponents of single isolated NACTs of the variousCIs. These values have to be obtained from ab initiotreatments; however, the entire field is formed byEq. (13).

To summarize our findings so far, we may saythat if indeed the radial component of a single CIcan be assumed to be zero we may present, almostfully analytically, the 2-D “field” of the NACTs fora two-state system formed by several CIs located atarbitrary points. Thus, Eq. (13) can be considered asthe nonadiabatic coupling field in the case of twostates.

In the next section, this derivation is finally ex-tended to a three-state system.

Three-State Case

DERIVATION OF THE EQUATIONS

To study the three-state case, we consider twoNACTs: one between the lowest and the interme-diate states, designated as �12 with its origin locatedat Pa(qa, �a), and the other between the intermediateand highest states, designated as �23 with its originlocated at Pb(qb, �b). As will be seen, in addition to�12 and �23 we have also to consider �13 although nodegeneracy point exists between the lowest andhighest states (unless all three states become degen-erate at the same point—a situation excluded here).

In other words, we shall show how the interactionbetween �12 and �23 forms �13, which does not havean origin of its own. Thus, the �-matrix for the mostgeneral case has to be of the form

� � � 0 �12 �13

��12 0 �23

��13 ��23 0�. (14)

The Curl equation for three (or more) states takesthe form [14, 15]

Curl� � �� 0. (15)

The derivation of the �-matrix elements will bedone in two steps: (1) by considering each of the CIsas being isolated, namely, the one independent ofthe other; and (2) refer to Eq. (15) for treating thetwo CIs as one complete system. Thus, within thefirst step we obtain zeroth-order expressions for �12and �23, i.e., �012 and �023, respectively, whereaswithin the second step we not only correct theseexpressions so that Eq. (15) is fulfilled for threestates but also derive the missing components ofthe �13 term. The study is done, as before, for aplane in CS employing polar coordinates.

To study the two isolated CIs, we have to treattwo-state Curl equations given in Eq. (1). Here, thefirst 2 � 2 �-matrix contains the (vectorial) element,i.e., �012 and the second 2 � 2 �-matrix contains �023.As before, each of the NACTs, �012 and �023, has thefollowing components:

�0jj�1 � ��0qjj�1, �0�jj�1�; j � 1, 2, (16)

where �0qjj�1 and �0�jj�1, j � 1, 2, were derived inthe previous chapter [see Eqs. (10)] and therefore nofurther treatment is necessary.

Recently, we discussed situations (based on abinitio calculations) where the two NACTs �12 and�23 overlap each other only slightly [8, 11]. Based onab initio calculations (as were carried out for theC2H molecule), it was found that in many cases theNACTs are not evenly distributed around their CIpoint but, rather, are concentrated along radialridges that start at the CI point (see Figs. 2 and 3).Thus, in case the two ridges are more or less par-allel, only slight overlaps are expected between thespatial distributions of the two CIs, in particularwhen the two CI points Px(qx, �x); x � a, b arelocated far enough from each other.



Thus, if �jj�1—the perturbed (vectorial) NACT—and the unperturbed NACT, �0jj�1, are related toeach other as

�jj�1 � �0jj�1 ��jj�1; j � 1, 2 (17)

then it follows, from the above-mentioned com-ments, that the components of the two vectorialperturbations, i.e., ��qjj�1 and ��jj�1, are likely to be(much) smaller than the corresponding compo-nents, namely, �0qjj�1 and �0�jj�1.

Next, we return to Eq. (15) and recall that we areinterested only in the components of �jj�1, j � 1, 2,in a plane perpendicular to the z-axis. It can beshown that if �0jj�1, j � 1, 2, do not possess az-component, the same applies to the perturbations��jj�1, j � 1, 2, as well as to �13.

Substituting Eq. (17) in Eq. (14) and the result inEq. (15) yields the following (inhomogeneous) dif-ferential equations for the components of ��jj�1, j �1, 2:

Curl��12� ��q12�

��

��12�

�q � ��13�0q23 � �q13�0�23

Curl��23� ��q23�

��

��23�

�q � �q13�0�12 � ��13�0q12,

(18)

where second-order terms were deleted. In this der-ivation, we employed the fact that

Curl�012 � Curl�023 � 0. (19)

In the same way, with similar assumptions, weobtain the (inhomogeneous) differential equationfor the components of �13:

Curl�13 ��q13

��

��13

�q � �0�12�0q23 � �0q12�0�23. (20)

Equation (20) is the an explicit Curl equation for acoupling that does not has a source of its own but isformed due to the interaction between two “phys-ical” CIs.

One of the more interesting feature of Eq. (20) isthat although �13 is not formed from conical inter-section this does not mean that its components areregular at every point. Within our model, namely,the situation where the radial component of eachindividual CI is identically zero, �13 is expected tobe regular everywhere. However, it could be that inrealistic cases, where these radial components arenot necessarily zero, Eqs. (20) may produce a loga-rithmic singular radial component for �13. Thus,according to Eqs. (20), or its extensions in the caseof subspaces containing more states, the effect ofthis new type of (logarithmic) singularities may

FIGURE 2. Equilines for the nonadiabatic couplingterms t12(x, y) and t23(x, y) as calculated for the C2Hmolecule for a fixed CC distance, i.e., rCC � 1.35 Å. (a)Equinonadiabatic coupling terms lines for the t12(x, y).(b) Equinonadiabatic coupling terms lines for t23(x, y).


1582 VOL. 90, NO. 6

have consequences regarding the physical featuresof molecular systems.

SENSITIVITY ANALYSIS

Equations (18) and (20) together with Eqs. (19)form the basic set of equations that enable the cal-culation of the nonadiabatic coupling matrix. Asnoted, this set of equations creates a hierarchy ofapproximations starting with the assumption thatcross-products on the right side of Eq. (20) havesmall values because at any point in CS at least oneof the multipliers in the product is small. To get anestimate, we assume that any of the components of�0jj�1, j � 1, 2, behaves like �aj, where aj, j � 1, 2,fulfill the following requirements:

0 � aj � 1; j � 1, 2 (21)

and

a1 a2 1. (22)

It is expected that aj approaches the value of 1 [sothat ��0jj�1� O(�)] when the point P(q, � ) is outsideits own ridge (formed by the �0jj�1CI/PI), but be-comes close to zero when P(q, � ) is located in itsown (i.e., in the jth ridge). Therefore, as can be seenfrom Eq. (20), the value of ��13� is O(�) when thepoint P(q, � ) is located at the vicinity of one of theridges but, otherwise, will be O(�2). This analysisshows that immaterial where the point P(q, � ) islocated the values of ��13� are about “one order ofmagnitude” smaller than those of ��0jj�1�.

The situation is less clear with regard to the twocorrection terms ��0jj�1 as presented in Eq. (18). Asan example, we discuss the equation for the com-ponents of ��012 [the first equation in Eqs. (18)]. It isnoted that the inhomogeneous term formed by thecross-product between the components of ��023 andthose of �13 will produce a correction O��1�a2��.Because the value of a2 varies from one region toanother, we have to analyze the three regions (seeFig. 2): (1) For points located at the (2, 3) ridge thevalue of a2 is close to zero. As a result, the correctionto the �012 components is of O(�), which is the sameorder of magnitude as those of the �012 componentsthemselves. (2) For points located outside the tworidges we have a2 1 and so ��012� is O(�2) butthe values of ��012� in this region are O(�) and sothe relative correction is of O(�). (3) For pointslocated at the (1, 2) ridge, a2 1 but because ��012�

O(0) the relative magnitude of the corrections be-comes O(�2).

The conclusion of this discussion is that in mostcases Eqs. (18) and (20) can be considered as therelevant extension of the two-state case to the three-state case. This extension is formed by solving asystem of decoupled first-order inhomogeneousdifferential equations. However, there are regionswhere the corrections obtained from these pertur-bative-type equations may not be good enough. Inthis case, these equations can be considered as ageneral presentation for an iterative procedure.

One of the more interesting outcomes revealedin this treatment is the way �13 is formed. Wefrequently claimed that two nonsuccessive statescannot form CI [10, 24] and so the natural questionarises, namely, whether �13 is identically zero ornot. If not, then the question is: What is the origin ofits existence? Our derivation shows that �13 is, un-ambiguously, formed via the interaction between�12 and �23 [see Eq. (20)].

The main difficulty with Eqs. (18) and (20) is thateach contains two unknown functions, which im-plies that these equations, although being inhomo-geneous, have numerous solutions and these can beuniquely determined only by introducing a mea-sure or a gauge. The possible solutions will bediscussed elsewhere.

Conclusions

In this article, we extended the idea [18, 19] toderive the nonadiabatic coupling terms by solvingthe Curl equations for given boundary conditions.Whereas in the two above-mentioned publicationswe considered the Curl equations for a two-statesystem, this treatment is now extended to a three-

FIGURE 3. Geometric positions (with respect to theCC axis) of t12(x, y) and t23(x, y) (see Fig. 2). Xij desig-nates the conical intersection formed by the ith and jthstates. All distances are in Å: (a) The (1, 2) conical inter-section; (b) the (2, 3) conical intersection.



state case where the first and second states formone CI and the second and third states form an-other. The spatial distribution of the two CIs areassumed to be overlapping only to a limited extentas, for instance, is the case shown in Figure (2) forthe three lowest states of the C2H molecule.

The extension from a two-state to a three-statesystem is not simply technical (increasing the num-ber of equations that have to be solved) because inthis process the Curl equation acquires newterms—nonlinear terms—that do not show up inthe two-state equations [compare Eq. (1) with Eq.(15)]. As is usually the case, nonlinear terms in-crease significantly the difficulties in solving thedifferential equations. In the present study, we de-veloped a perturbative scheme, which “linearizes”the equations. In a sensitivity analysis performedfollowing the derivation, it was shown that thisscheme guarantees convergence if the overlap be-tween the two interacting CIs is not too strong.

One of the more interesting outcomes of thisstudy is the revealing of the process that leads tothe construction of �13, the NACT that couples thefirst and third states. In numerous publications, weclaimed that CIs are not allowed unless they areformed between successive states. Thus, a CI be-tween the first and third states cannot be formed.From ab initio calculations, it is known that �13 issmall but not absolutely zero [25]. In the presentstudy, it is shown for the first time that �13 isformed as a result of the interaction between �12

and �23 and, to be more specific, the components ofCurl�13 are equal to cross-products between thecomponents of �12 and �23.

As a final point, because �13 is formed by �12 and�23 it may also acquire some of their more charac-teristic features, namely, becoming nonregular(having a logarithmic singularity) at the singularitypoints of �12 and �23.

ACKNOWLEDGMENTS

M.B. thanks Professor G. D. Billing for his kindhospitality during his stay at the Department ofChemistry, the University of Copenhagen, and theDanish Research Training Council for partly sup-porting this research. Also, he thanks Dr. Roi Baerfrom the Department of Chemistry at the HebrewUniversity in Jerusalem for illuminating discus-sions.

Appendix: Treatment of a SeamRemoved from the Origin ofCoordinates

We start by writing the Curl equation in Eq. (3)for a vector �(x, y) in Cartesian coordinates:

��x

� y ��y

� x � 0. (A1)

The solution to Eq. (A1) is

��x, y� � f�yx� �yix xiy

x2 y2 , (A2)

where ix and iy are unit vectors along the x and yaxes, respectively. To shift this solution from theorigin to some given point (xj0, yj0), the variable x isreplaced by (x � xj0) and the variable y by (y � yj0)so that the solution of Eq. (A1) is given in the form

��x, y� � f�y � yj0

x � xj0� ��y � yj0�ix �x � xj0�iy

�x � xj0�2 �y � yj0�

2 .

(A3)

Next, we are interested in expressing this equationin terms of polar coordinates (q, � ). For this pur-pose, we recall the following relations:

x � q cos �; y � q sin � (A4)

and introduce the following definitions:

x � xj0 � qjcos �j; y � yj0 � qjsin �j. (A5)

Because we are interested in the polar componentsof �(q, � ), i.e., �q and ��, we need also to know theirrelation with �x and �y, which was derived sometime ago [26]:

�q � ��1 �

�q�2 � cos ��x sin ��y

�� 1 �

��2 � q��sin ��x cos ��y�, (A6)

where �1 and �2 are the two lowest electronic adi-abatic wave functions. Employing Eqs. (A3), (A5),and (A6), we finally get


1584 VOL. 90, NO. 6

�q�q, � � � �f��j�1qj

sin�� j�

��q, � � �qqj

f ��j�cos�� j�. (A7)

Equations (A7) are the equations employed in thetext [see Eqs. (11)].

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curl equations as substratum for the derivation of the electronic nonadiabatic coupling terms

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