A rigorous approach to the formulation of extended Born-Oppenheimer equation for a three-state...

A Rigorous Approach to theFormulation of Extended Born-Oppenheimer Equation for a Three-State System

BIPLAB SARKAR,1,2 SATRAJIT ADHIKARI1,3

1Department of Chemistry, Indian Institute of Technology Guwahati, North Guwahati,Guwahati-781 039, India2Department of Chemistry, North-Eastern Hill University, Shillong-793 022, India3Department of Physical Chemistry, Indian Association for the Cultivation of Science, Jadavpur,Kolkata-700 032, India

Received 3 April 2008; accepted 30 June 2008Published online 15 October 2008 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.21870

ABSTRACT: If a coupled three-state electronic manifold forms a sub-Hilbert space, itis possible to express the non-adiabatic coupling (NAC) elements in terms of adiabatic–diabatic transformation (ADT) angles. Consequently, we demonstrate: (a) Those explicitforms of the NAC terms satisfy the Curl conditions with non-zero Divergences; (b) Theformulation of extended Born-Oppenheimer (EBO) equation for any three-state BOsystem is possible only when there exists coordinate independent ratio of the gradientsfor each pair of ADT angles leading to zero Curls at and around the conicalintersection(s). With these analytic advancements, we formulate a rigorous EBOequation and explore its validity as well as necessity with respect to the approximateone (Sarkar and Adhikari, J Chem Phys 2006, 124, 074101) by performing numericalcalculations on two different models constructed with different chosen forms of theNAC elements. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem 109: 650–667, 2009

Key words: Born-Oppenheimer treatment; Non-adiabatic coupling elements; ADT;Curl-Divergence equation; extended Born-Oppenheimer equation

Correspondence to: S. Adhikari; e-mail: [email protected] grant sponsor: Department of Science and Technol-

ogy (DST, Government of India).Contract grant number: SP/S1/H-53/01.Contract grant sponsor: UGC, Government of India through

DSA-SAP–III.

International Journal of Quantum Chemistry, Vol 109, 650–667 (2009)© 2008 Wiley Periodicals, Inc.

Introduction

H erzberg and Longuet-Higgins’ (HLH)[1] treat-ment on the Jahn-Teller model lead to an inter-

esting observation—the real valued electronic wave-function of that model undergoes a sign change as thenuclear coordinates make a closed path around theconical intersection (CI) and the Born-Oppenheimer(BO)[2] treatment breaks down. Though HLH cor-rected this deficiency by multiplying the double-val-ued eigenfunctions with a complex phase factor (com-monly termed as Longuet-Higgins’ phase) such thatthe resulting wavefunctions become single valued, itis important to emphasize that Longuet-Higgins’“modification” of the electronic eigenfunctions is im-posed in an ad hoc manner. Mead and Truhlar [3]introduced a vector potential in the nuclear Hamilto-nian to generalize the BO equation in order to accountthe so-called geometric phase (GP) and such ap-proach is a reminiscent of the complex phase factortreatment of HLH. Both these theoretical approachpredicted that the GP effect could leave its’ signatureon scattering processes. Kuppermann et al. [4] calcu-lated integral and differential scattering cross sectionsof H3 isotopic system and demonstrated [5] the effectof GP on reactive/nonreactive transition probabilities.Moreover, the Extended Born-Oppenheimer (EBO)equations [6] by considering GP based vector poten-tials were being used to perform scattering calcula-tions on tri- and tetra-atomic reactive systems. Suchtheoretical predictions and calculations demand toexplore the origin of GP from first principle.

Any first principle based theory starting with BOtreatment considers the fact that slow-moving nu-clei is distinguishable from fast-moving electrons inmolecular systems. This distinction helps to imposethe BO approximation by neglecting the effect ofupper electronic state(s) on the lower, that is, thenon-adiabatic coupling (NAC) elements are as-sumed to be negligibly small. Though the implica-tion of this approximation is independent of theeigenspectrum of the system, the ordinary BOequation is frequently used for calculations even forsystems with large NAC terms based on the as-sumption that upper electronic states are “closed”to the ground at low energies. On the other hand,even if the projection(s) of the total wavefunctionon the upper electronic state(s) are negligibly smallat enough low energies, their product with largeNAC terms leading to the effective coupling amongthe states could be finite in magnitude and thereby,the BO approximation could break down for such

situations. Therefore, one needs to pursue theoret-ical development in such a way that the beyondBorn-Oppenheimer effects are included in the dy-namical calculations. While developing such theo-ries, Mead and Truhlar [7] mentioned that the con-sideration of the entire Hilbert space (n � N) toincorporate the couplings among the electronicstates is indeed a trivial approach to demonstrate.In the same article [7], they predicted that for anyrealistic description of the electronic wavefunction,the Curl of the non-adiabatic coupling does notvanish. A general vector field can be decomposedinto longitudinal and transverse components,where the longitudinal component can be ex-pressed as a derivative of a scalar and the trans-verse component by the curl of a vector. The ADTcan at best remove the longitudinal component ofthe derivative coupling. The longitudinal and trans-verse components are referred to as the removableand nonremovable couplings.

The general characteristics of the removable andnonremovable components have been discussed byKendrick et al. [8]. When the energy eigenvalues arewell separated, the removable and nonremovablecouplings will be of the same order. At sufficientlylow energies (well below the energy of the upperstate), these coupling can be ignored in dynamicscalculations due to the 1/M prefactor. At the closeproximity of a degeneracy, only the removable cou-pling is singular and according to the degenerateperturbation theory, the nonremovable couplingsare insignificant [9]. It means that the ADT anglecan be obtained by integrating the derivative cou-pling at and around the same region. On the con-trary, away from the CI, the contribution from thenonremovable coupling appears in path dependentintegrals for the ADT angles and therefore, closedline integrals of the derivative coupling [10] will notbe multiples of �. The inclusion of more electronicstates can reduce this problem [11], however,greatly increases the computational cost of ab initioquantum chemistry and dynamical calculations.

One can separate the removable and nonremov-able couplings by solving Poisson’s equation for theADT angle � [12]. Since there are many possibledefinitions for the boundary conditions on � [13],there is no unique solution. Moreover, the solutionof Poisson’s equation is computationally too expen-sive to be carried out for molecules of more thanthree atoms. Since the Born-Oppenheimer approx-imation implies that it is not necessary to find thebest diabatic basis, one can find a diabatic basis forwhich the residual couplings can be neglected and

BORN-OPPENHEIMER EQUATION FOR A THREE-STATE SYSTEM

VOL. 109, NO. 4 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 651

such bases are referred to as quasidiabatic bases. Therequirements for a quasidiabatic basis are easier tosatisfy: (a) The singularity in the derivative couplingmust be transformed away; (b) The residual couplingsmust be negligible. It is desirable for a diabatic basis toestimate the residual couplings to ensure that no spu-rious coupling has been incorporated. If it is neces-sary, the residual couplings could be perturbativelyincluded in scattering calculations.

On the contrary, while formulating the extendedBorn-Oppenheimer equation for a single surfacenuclear Schroedinger equation (SE), it is a matter ofcontemporary research how elegantly one can in-clude the effects of off-diagonal (so called non-adiabatic coupling) terms [14] to the diagonal. Con-sidering two coupled electronic states as sub-Hilbert space, Baer [15, 16] derived a new set of twocoupled BO equations by grafting the effects ofNAC terms into the diagonal and formulated thesingle surface approximate EBO equations by im-posing the condition, namely, the upper electronicstate is classically closed with respect to the groundstate. This EBO equation is being used to calculatetransition probabilities in a two-arrangement-chan-nel model [17, 18] and reproduces the correct sym-metry transitions as obtained from so called numer-ically exact diabatic SE. In an alternative attempt,Varandas and Xu [19] reformulated the two-stateadiabatic nuclear SE by casting the NAC elementsin terms of nuclear coordinate dependent electronicbasis functions angle (mixing angle), found the one-to-one correspondence between mixing [19] andadiabatic–diabatic transformation (ADT) [20] an-gles and thereby, derive the single surface EBOequation in the vicinity of degeneracy.

The first attempt to formulate EBO equation forany three-state coupled BO system in the adiabaticrepresentation of nuclear SE and to explore its’workability has been carried out by Baer et al. [21]and Adhikari et al. [22], respectively, considering amodel situation. Even though this derivation doesnot include the general features of any BO system,the formulation demonstrates the viability to deriveEBO equation and thereby, shows the scope forfurther theoretical development. Moreover, it pre-dicts that EBO equations can provide meaningfulsolution only when eigenvalues of NAC matrix aregauge-invariant. Sarkar et al. [23] has performed ageneralized BO treatment of any three coupled elec-tronic states and formulated an approximate EBOin terms of electronic basis functions angles. Thesame article also finds that calculated results ob-tained by using diabatic and approximate EBO

equations for various chosen forms of NAC ele-ments either have excellent or good agreement asthe eigenvalues of NAC matrix are either exactly orapproximately gauge invariant, respectively.

In this article, we reformulate the explicit formsof the non-adiabatic coupling elements along withtheir Curl-Divergence equations in terms of ADTangles by considering the validity of ADT conditionfor any three-state sub-Hilbert space. Since the nec-essary condition to derive the EBO equations is theexistence of a relation among the ADT angles im-plicating zero Curls (apart from Curl conditionsbeing satisfied) at least around the CIs, we brieflypresent the analytical proof for the validity of suchrelations considering nuclear coordinates depen-dent ADT angles. The major aim of this article is toformulate a rigorous EBO equation in terms of ADTangles and to explore it’s necessity with respect tothe approximate one [23]. In this context, we wishto construct two models by choosing various setson the form of the ADT angles vis-a-vis NAC termsand thereby, to find out the validity of the rigorousEBO equation with respect to the diabatic one andits’ advantage, if any, with respect to the approxi-mate EBO equation [23].

Theoretical Development on the BOTreatment of Sub-Hilbert Space andits Rigorous EBO Equation

The investigations on the validity/existence ofsub-Hilbert space requires a detailed discussion.When the derivative coupling is large, its’ nonre-movable component is relatively small, whereas ifthe derivative coupling is small, the nonremovablepart is a relatively significant component of thecoupling vector. Kuppermann and coworkers [13,24], Baer and coworkers [25, 26], and Yarkony andcoworkers [10, 27] carried out investigations on thisissue to demonstrate the possibility on the existenceof sub-Hilbert space. The nonremovable couplingshave been reported for the H3 system [13]. It wasobserved that the nonremovable couplings are atleast an order of magnitude lower than the deriva-tive coupling when the energy difference is lessthan 180 mH and the nonremovable coupling iscomparable to the derivative coupling when theenergy difference is greater than 180 mH. Baer et al.[25, 26] studied a tetra-atomic system, C2H2

�, toinvestigate the topological effect both for the two-state (Abelian) and multistate (non-Abelian) cases.

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652 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 109, NO. 4

In case of a tetra-atomic system, topological effectsare revealed when one atom surrounds the triatomaxis or when two atoms surround (at a time) thetwo atoms. In other words, it was shown that for atetra-atomic system not only a tri-atom axis buteven a two-atom axis forms a seam that containsdegeneracy points. For the treatment of non-adia-batic coupling terms, they distinguish between thecase where the NAC matrix is of 2 � 2 dimensionand the case where it is of the 3 � 3 dimension.Thus, the first case applies to the two-state Hilbertsubspace and the second to the three-state Hilbertsubspace. On the other hand, Yarkony [10] investi-gated the nonremovable part of the derivative cou-plings by considering the integral of derivative cou-pling along closed loops in the vicinity of the 12A�� 22A� seam of CIs in H3 system. It was noticed thatas the radial coordinate (�) tends to zero, the con-tribution of the nonremovable part decreases rap-idly and when � increases, the upper state ap-proaches the energy of manifold of Rydberg statessuch that the contribution from derivative cou-plings to this states becomes significant.

We demonstrate an alternative version of firstprinciples based BO treatment by assuming theexistence of any three-state electronic sub-Hilbertspace considering the presence of CI(s) anywhere inthe nuclear configuration space and formulate EBOequation in a rigorous manner. Since we assumethese three-states as either decoupled or approxi-mately decoupled from rest of the states of a mo-lecular system, the BO expansion of the wavefunc-tion for this subspace of the Hilbert space alongwith the total electron–nuclei Hamiltonian in theadiabatic representation are presented as

��n,e� � �i�1

3

�i�n��i�e,n�,

H � Tn � He�e,n�,

Tn � �2

2m�n

n2,

He�e,n��i�e,n� � ui�n��i�e,n�, (1)

where the eigenfunction ��i�e,n�� of the electronicHamiltonian, He�e,n�, is defined by the sets of nu-clear (n) and electronic (e) coordinates with nuclearcoordinate dependent eigenvalue, ui(n). Indeed, itis obvious to specify that Tn is the nuclear kinetic

energy (KE) operator and the expansion coefficient,�i(n), shall appear as nuclear wavefunction.

The time-independent Schroedinger equation,H��n,e� � E��n,e�, for the total electron–nuclearHamiltonian and the BO expansion of the sub-Hil-bert space molecular wavefunction [Eq. (1)] bringthe matrix representation of adiabatic nuclear SE

�j�1

3

�Hij � Eij��j�n� � 0, i � 1,2,3,

Hii � �2

2m�2 � 2� ii�1� � � � ii

�2�� ui�n�,

Hij � �2

2m�2� ij�1� � � � ii

�2�� Hji† ,

� ij�1� � �i�e,n�� j�e,n��, ij

�2� � �i�e,n��2�j�e,n��,

�i�e,n��j�e,n�� ij (2)

where �ij�1� and ij

�2� are the elements of non-adiabaticcoupling matrices of the first ��1�� and second ��2��kind, respectively. Moreover, it is straight forwardto show that for a given Hilbert space the matrices,��1� and �2� are related as

�2� � � �1� � � �1� � � � �1� (3)

and thereby, we can arrive [from Eqs. (2) and (3)]the following compact form of kinetically couplednuclear equations

�2

2m� � �12 �13

� �12 � �23

� �13 � �23 ��

2

��1

�2

�3�

� �u1 � E 0 00 u2 � E 00 0 u3 � E��

�1

�2

�3� � 0, (4)

where the NAC matrix �� 1�� is defined as,

� � � 0 �12 �13

� �12 0 �23

� �13 � �23 0� . (5)

Since the three-states constitute the sub-Hilbertspace (i.e., a complete space for the present case), itis possible to transform (� � A�d) the adiabaticnuclear SE [Eq. (4)] to the diabatic one as below,



��

2

2m2 � E 0 0

0 �2

2m2 � E 0

0 0 �2

2m2 � E��1

d

�2d

�3d�

� �W11 W12 W13

W21 W22 W23

W31 W32 W33

��1d

�2d

�3d� � 0, (6)

where W � A†UA with Uij � uiij, under the con-dition

� A � �A � 0. (7)

This equation was first formulated elsewhere[20], and is known as adiabatic–diabatic Transfor-mation (ADT) condition. To a obtain a meaningfulsolution of Eq. (7), we have to ensure that the

chosen form of A matrix has the following features:(a) It is orthogonal at any point in configurationspace; (b) Its’ elements are cyclic functions withrespect to a parameter, that is, starting with an unitdiagonal matrix, the chosen form of A matrix has togenerate a diagonal matrix with even number (�1)safter completing the cycle.

In case of three-dimensional Hilbert space, thereare nine elements in the ADT matrix (A). Since themodel form of A has to be an orthogonal matrix andthe orthonormality conditions demand the fulfill-ment of six relations, three independent variablesnamely Euler like angles of rotation [�12(n), �23 (n),and �13(n)], commonly called ADT angles, are thenatural requirement to construct the three-state Amatrix by taking the product of three 2 � 2 rotationmatrices, A12(�12), A23(�23), and A13(�13). Let us de-fine these three 2 � 2 rotation matrices [A12(�12),A23(�23), and A13(�13)] and one of the ways of theirproduct (A) as:

A��12,�23,�13� � A12��12� � A23��23� � A13��13� � � cos�12 sin�12 0� sin�12 cos�12 0

0 0 1��1 0 0

0 cos�23 sin�23

0 � sin�23 cos�23

�� cos�13 0 sin�13

0 1 0� sin�13 0 cos�13

� � �cos�12cos�13 sin�12cos�23 cos�12sin�13

� sin�12sin�13sin�23 � sin�12cos�13sin�23

� sin�12cos�13 cos�12cos�23 � sin�12sin�13

� cos�12sin�13sin�23 � cos�12cos�13sin�23

� sin�13cos�23 � sin�23 cos�13cos�23

� (8)

When we substitute the above model form of Amatrix [Eq. (8)] and the antisymmetric form of ma-trix [Eq. (5)] in Eq. (7), the simple manipulation asperformed by Top and Baer [28] and Alijah and Baer[29] leads to the following equations for ADT angles:

� �12 � � �12 � tan�23��13cos�12 � �23sin�12�, (9a)

� �23 � � ��13sin�12 � �23cos�12�, (9b)

� �13 � �1

cos�23��13cos�12 � �23sin�12�, (9c)

which in turn brings the explicit form of matrixelements in terms of ADT angles

�12 � � � �12 � sin�23� �13, (10a)

�23 � sin�12cos�23� �13 � cos�12� �23, (10b)

�13 � �cos�12cos�23� �13 � sin�12� �23. (10c)

Once the non-adiabatic coupling elements �12, �23,and �13 are evaluated by using ab initio calculationfor a particular nuclear configuration, the solutionof Eqs. (9) provides the ADT angles for the samenuclear configuration. On the other hand, if wehave the total electron–nuclear Hamiltonian of amolecular system in the diabatic representation,one can calculate the ADT matrix by diagonalizingthe W matrix [Eq. (6)] and thereby, obtain the NACelements through Eq. (7).

A Curl condition [20] for each NAC element, �ij,has been derived and proved to exist for an isolatedgroup of states (sub-Hilbert space) by consideringthe analyticity of the ADT matrix A for a pair ofnuclear degrees of freedom,

pijq �

qijp � �qp�ij � �pq�ij,

ijp � �i�p�j�, ij

q � �i�q�j� (11)

SARKAR AND ADHIKARI


where the Curl due to vector product of NAC ele-ments and the analyticity of ADT matrix are givenby Cij � �qp�ij � �pq�ij and Zij � / pij

q

� / qijp, respectively in terms of Cartesian coor-

dinates p and q with p � / p and q � / q.Thus, the explicit form of Curl equation in terms

of ADT angles for each NAC element is obtained byusing Eqs. (10) and (11) as below

Curl 12pq � C12 � Z12 � � cos�23�q�23p�13

� p�23q�13� (12a)

Curl 23pq � C23 � Z23 � cos�12cos�23�q�12p�13

� p�12q�13�

� sin�12sin�23�q�23p�13 � p�23q�13�

� sin�12�q�12p�23 � p�12q�23� (12b)

Curl 13pq � C13 � Z13 � sin�12cos�23�q�12p�13

� p�12q�13� � cos�12sin�23�q�23p�13 � p�23q�13�

� cos�12�q�12p�23 � p�12q�23� (12c)

where the divergence of �ijs [Eq. (10)] are given by

div �12 � 2sin�12cos�12cos2�23�� 13 � � �13�

� 2sin�12cos�12�� 23 � � �23�

� 3cos2�12cos�23�� 13 � � �23�

� sin2�12cos�23�� 13 � � �23� � sin�232�13 � 2�12

(13a)

div �23 � 2sin�12sin�23cos�23�� 13 � � �13�

� 3cos�12cos�23�� 12 � � �23� � 3sin�12�� 12 � �23�

� sin�12sin�23�� 13 � � �23� � sin�12cos�23�12

� cos�122�23 (13b)

div �13 � 2sin�12sin�23cos�23�� 12 � � �12�

� 3sin�12cos�23�� 12 � � �13� � 3cos�12�� 12 � � �23�

� cos�12sin�23�� 13 � � �23� � cos�12cos�232�13

� sin�122�23 (13c)

Moreover, it is possible to show that there arealtogether six (6) different ways to take the productof the three rotation matrices [A12(�12), A13(�13), andA23(�23)] to obtain the ADT matrix (A)

A � Pn�A12��12� � A23��23� � A13��13��, n � 1,. . .N!,

(14)

where Pn is the nth permutation between two rota-tion matrix. Indeed, it is important to note that eachADT matrix can provide similar set of differentialequations for ADT angles [Eq. (9)], NAC elements[Eq. (10)] and their Curl-Divergence equations [Eqs.(12) and (13)].

Since � �ijs and in general, 2�ijs are non-zeroaround the CI, the divergence of the vector field ��ij�are nonvanishing for any arbitrary values of ADTangles and thereby, the vector field may show upnon-zero Curl [30, 31] also. When a non-adiabaticcoupling term of the kind, �ii�1�n�, is associated witha singularity (pole) at the (i,i � 1) CI point, ab initiocalculations [32] demonstrate that NAC term de-cays like 1/r, where r is the distance from the CI.Such vector field could be resolved into irrotational(longitudinal) and solenoidal (transverse) compo-nents [13, 24, 30, 31], where, by definition, the Curlof longitudinal part is zero but Curl of transversepart may or may not. The Abelian (commuting) andnon-Abelian (noncommuting) magnitudes [33] ofCurl equations are the key issue to the formulationof EBO equation. For any two-state (N � 2) sub-Hilbert space, the components of NAC matricessatisfy the Abelian Curl equation but for N � 3cases, the Abelian or non-Abelian nature of Curlequation has been explored very recently.

The following section demonstrates that in orderto formulate single surface EBO equation, why it isnecessary to find out the nature of Curl ij

pqs quan-titatively, at least around the point of CI, for a giventhree-state sub-Hilbert space. Let us start with thematrix representation of three-state adiabaticSchroedinger equation [Eq. (4)] as given by,

�2

2m�� 2� � �U � E�� 0, Uij � uiij. (15)

We wish to pursue an unitary transformation onEq. (15) by a matrix, G�� G��, to bring the effectof off-diagonal elements (NAC terms) on the diag-onal such that it leads to the following form:

�2

2m�G†� G � i�� 2� � �V � E�� 0,

V � G†UG, i�� G†�G, (16)



where the NAC matrix, �, need to hold an importantfeature, namely, its’ eigenvalues (0 and � i�� ) arevectors (including the null vector) to ensure physi-cally meaningful (a scalar) Hamiltonian [Eq. (16)] andthereafter, one can impose the BO approximation,��1� � � ��i�, i � 2, 3, by considering the upperelectronic states as classically closed at low enoughenergy, to formulate the single surface adiabatic nu-

clear SE (EBO) [21]. Since the requirement of Eq. (16)dictates that the eigenvalues ( � i�� ) of � matrix mustbe vectors, and this can be achieved only when thecomponents of � matrix commutes (Abelian) witheach other, the � matrix could be written as the prod-uct of a vector function, �� 12 or �23 or �13� and aADT angle dependent antisymmetric scalar matrix,g(�12, �23, �13) as given below

� � � �12�0 � 1 � sin�23�p�13

p�12� � sin�12�p�23

p�12� � cos�12cos�23�p�13

p�12�

1 � sin�23�p�13

p�12� 0 � cos�12�p�23

p�12� � sin�12cos�23�p�13

p�12�

sin�12�p�23

p�12� � cos�12cos�23�p�13

p�12� cos�12�p�23

p�12)�sin�12cos�23�p�13

p�12� 0

�� 12 � g��12,�23,�13�, (17)

with eigenvalues, 0 and � i�� , where

�� 12�1 � �p�13

p�12� 2

� �p�23

p�12� 2

� 2sin�23�p�13

p�12�

12. (18)

In other words, this product form of NAC ma-trix demands the validity of identities, (p�13/p�12) � (q�13/q�12/), (p�23/p�12) � (p�23/p�12), and (p�23/p�13) � (q�23/p�13), namely,Curl 12

pq � 0s [see Eq. (12)]. The following sectionexplores whether the above identities are valid ornot, that is, Curl ij

pqs are zero or not at and aroundCI(s). We explore the nature of these identities byusing the Jacobian determinant (J(r, �, �)) defined atand around the point of CI for the transformationfrom Cartesian to polar as given by

J�r,�,�� x r

y r

z r

x �

y �

z �

x �

y �

z �

� � sin � cos � sin � sin � cos �

r cos � cos � r cos � sin � � r sin �� r sin � sin � r sin � cos � 0

�.(19)

When the origin of the coordinate system [r � 0(x � 0, y � 0, z � 0)] coincides with the point of CIor even if the point of CI(s) is away from the originof the coordinate system, parametric representationfor the vector equation of a conical surface predictsJ(r, �) � 0, J(r, �) � 0, and J(�, �) � 0 at thesingularity (CI) (see Appendix B of Ref. [23].) lead-ing to the quantities ( �23/ y �13/ x � �23/ x �13/ y), ( �23/ z �13/ x � �23/ x �13/ z), and( �23/ y �13/ z � �23/ z �13/ y), vis-a-vis Curl12

pq(p, q � x, y, z) or ( �23/ � �13/ r � �23/ r �13/ �), ( �23/ � �13/ r � �23/ r �13/ �), and( �23/ � �13/ � � �23/ � �13/ �), vis-a-vis Curl12

pq(p, q � r, �, �) are either identically or approxi-mately zero at and around the CI.

With these implications on Curl equations, weintend to rewrite Eq. (16) as,

�2

2m�� i�� 2�� 2

2m��G†� 2G� � 2��

� �i�� G†� G� � i�� G†� Gi�� i��

� ��V � E�� 0, (20)

where G is the transformation matrix that diago-nalizes the antisymmetric scaler matrix, g(�12, �23,�13) [Eq. (17)] instead of NAC matrix, � (�12, �23, �13)[Eq. (15)]. For symbolic convenience, now onwardswe shall replace G† as Gd and its’ element �G†�ij as�Gd�ij� � Gij

d�.The ith BO equation can be written from the

matrix equation [Eq. (20)] as below,

SARKAR AND ADHIKARI


�2

2m�� i�� 2��i �2

2m��k

Gikd 2�k � �

k

2�Gikd �k�

� �k

i�� iGikd � �k

� �k

i�� i� �Gikd �k� � �

km

Gikd � ��km�m�

� �km

� �Gikd �km�m�� V � E��i � 0. (21)

We manipulate the Eq. (21) by considering the fol-lowing aspects: (a) Since the matrix representationof the ADT (� � Gd�) is given by

��1

�2

�3

� � �g3

��

g2

�

g1

�

�g1g3 � i�g2

�2��

g1g2 � i�g3

�2��

�

�2�� g1g3 � i�g2

�2��

g1g2 � i�g3

�2��

�

�2�

��1

�2

�3

� ,

(22)

with

g1 � �1 � sin�23�p�13

p�12�,

g2 � � sin�12�p�23

p�12� � cos �12 cos �23�p�13

p�12�,

g3 � � cos �12�p�23

p�12� � sin�12 cos �23�p�13

p�12�,

� � �g22 � g3

2 � ��p�23

p�12� 2

� cos2�23�p�13

p�12�2

12,

� � �g12 � g2

2 � g32 � �1 � �p�13

p�12� 2

� �p�23

p�12� 2

� 2sin�23�p�13

p�12�

12,

one can have the general identity

�k �1

Gkkd �k � �

l�k

Gkld

Gkkd �l k,l � 1,2,3. (23)

(b) The product, V�, for the ith equation can berearranged as below,

�V��i � u1�i � �j�2

3

Gijd�uj � u1��j, i � 1,2,3. (24)

Finally, we substitute Eqs. (23) and (24) for i � 1in the Eq. (21) and impose the BO approximation,��1� � � ��i�, i � 2, 3 (at low enough energy, boththe upper electronic states are classically closed) toobtain the ground state EBO equation as,

�2

2m�� i�� 1�2�1 �

2

2m� � 2�� G11d

G11d �� 1

� 2�� G11d

G11d � 2

�1 � �2G11d

G11d ��1 � i�� 1�� G11

d

G11d ��1

� �� G12d

G11d ��21�1 � �� G13

d

G11d ��31�1 � �u1 � E��1 � 0.

(25)

If we now introduce the approximationnamely the transformation matrix G elements areslowly varying functions of nuclear coordinatesand thereby, the matrix (G) commutes with thegradient operator � , Eq. (25) lead to the followingapproximate EBO Eq. (23) for the ground elec-tronic state

�2

2m�� i�� 1�2�1 � �u1 � E��1 � 0, (26)

where this equation [Eq. (26)] with simple BO ap-proximation becomes

�2

2m2�1 � �u1 � E��1 � 0. (27)

In the following sections, the validity of therigorously formulated EBO equation [Eq. (25)]has been justified by comparing the numericallycalculated transition probabilities on the groundelectronic state with the exact results obtainedfrom the diabatic equation [Eq. (6)], where in bothcases, calculations are performed on two modelsinvolved with strong non-adiabatic effects. Thecalculated results obtained by using approximateEBO equation [Eq. (26)] and BO approximateequation [Eq. (27)] are also presented for compar-ison with diabatic [Eq. (6)] and rigorous EBO [Eq.(25)] results.



The Models and the Non-AdiabaticCoupling Elements

The non-adiabatic coupling among any threeelectronic states could show up single or multiplenumber of CI(s) at a particular or different nu-clear configuration(s) space, respectively. One ofthe commonest possibilities is the presence of twoCIs, namely, the first and second electronic statesmay indicate a CI at a particular nuclear config-uration and the second and third electronic statecan have another CI at different nuclear configu-ration. There is another possibility, not really un-common—the three electronic states could showup the CI at a single point. When the CIs are atdifferent nuclear configuration space, they couldbe far apart from each other or even close enough.If they are far away from each other, effectivelythe 3 � 3 BO system translates into two 2 � 2 BOsystems. In such situation, the matrix G, whichdiagonalizes the anti-symmetric scalar matrix g,commutes with the operator � and thereby, Eq.(26) could be considered as the rigorous EBOequation. On the other hand, when the systemhas two close enough CIs or a single CI amongthe three-states, the G matrix does not commutewith � and Eq. (25) would be the rigorous form ofthe EBO equation.

The Model A

THE NON-ADIABATIC COUPLING ELEMENTSAND ADIABATIC PESS

The non - adiabatic coupling elements of theC2H molecule is a good example to cite in con-nection with our following proposed Model Awith some differences also. The ab initio calcula-tions [34, 35] for this molecule show that there isa CI between 2 2A� and 3 2A� states at a particularnuclear configuration and are two CIs between 32A� and 4 2A� states at different configuration ofthe nuclear space, where the spatial distributionsof the NAC terms, �12, �13, and �23 depending uponthe size of the circular contours dictate [34]whether the three-state problem can be resolvedinto two approximate two-state systems or not.There are numerous molecules with similar non-adiabatic coupling profiles among the three con-secutive adiabatic electronic states.

The construction of Model A (a 3 � 3 BOsystem) is such that it breaks up into two 2 � 2BO systems when the point of CI between the firstand the second states is far away from the pointof CI between the second and the third states.This become possible by introducing �12 and �23 ina way that their spatial distributions either do nothave or have very small overlap with respect tothe radial coordinate at the asymptote, where �13

is made nearly or identically zero. Such func-tional form of non-adiabatic terms can be incor-porated by manipulating the nuclear coordinate(let say x and y) dependence of the ADT angles,�12(n), �23(n), and �13 (n). Moreover, since weassume that the Model A do not have any CIbetween the first and third state for the entirerange, �� x, y � �, the ADT angle (�13)between those states is expected to be small. Evenif the ADT angle �13(n) between these two-statesis assumed either remain constant or zero for theentire nuclear configuration space, still it does notmean that the non-adiabatic coupling term, �13, isexactly zero since it can grow due to the spatialoverlapping of the NAC terms, �12 and �23 [see Eq.(10)]. To fulfill the above expectations fromModel A, we introduce the following choice ofspatial distribution on the ADT angles: �12(x, y) �1/2 tan�1( y/x�x0) � 1/4 sin[2 tan�1( y/x �x0)], �23 (x, y) � 1/2 tan�1( y/x0 � x) � 1/4 sin[2tan�1( y/x0�x)], and �13(x, y) � 0.

It may be noted that the choice of the func-tional form of these three ADT angles are notquite arbitrary, particularly, they are the fittedones from ab initio calculated data [34]. We sub-stitute those angles in the Eq. (10) to obtain thespatial distributions of non-adiabatic coupling el-ements and display the corresponding NAC ele-ments in Figures l(a)–(c) for various separation[2x0 (� 4, 2, 0.5 Å)] between the CIs. The rightinset in Figure l(a) shows the profile of ADTangles (�12/�23) as function of the nuclear coordi-nate �[�tan�1�y/x)], where the left inset in Fig-ures l(a)–(c) present how the functional form of�13� grows as �12� and �23� appear closer andcloser. For a given separation of CIs (2x0), theabove functional form of NAC terms along withthe following chosen form of adiabatic PESs [17,18],

u1� x,y� �12��0 � �1� x��2y2 � A1 � f1� x,y�,

SARKAR AND ADHIKARI


u2� x,y� �12��0

2y2 � �D1 � A1� � f1� x,y� � A1

� f2� x,y� � D1,

u3� x,y� �12��0

2y2 � �D1 � A1� � f2� x,y� � D2,

�1� x� � �1exp� ��x � x0�

2

�12 ,

f1� x,y� � exp� ��x � x0�

2 � y2

�2 ,

f2� x,y� � exp� ��x � x0�

2 � y2

�2 , (28)

defines the corresponding adiabatic nuclear SE [Eq.(4)]. Figure 2(a) presents those adiabatic PESs for2x0 � 4 Å with the potential parameters as given by� � 0.58 amu, A1 � 3.0 eV, D1 � 5.0 eV, D2 � 10.0eV, �0 � 39.14 � 1013 s�1, � 1 � 7.83 � 1013 s�1, �� 0.3 Å and �1 � 0.75 Å, respectively.

THE DIABATIC AND RIGOROUS EBOEQUATION

The ADT angles as chosen in section the non-adiabatic coupling elements and adiabatic PESs arebeing used to construct the ADT matrix [Eq. (8)]and thereby, to transform the adiabatic to diabaticSE [Eq. (6)]. The detailed expression of adiabaticand diabatic SEs for any generalized form of ADT(mixing) angles were presented elsewhere (see Ap-pendix A of Ref. [23]). On the other hand, in orderto formulate the EBO equation for Model A, wemake use the following form of the matrix byconsidering �13 � 0 in the Eq. (17)

� � ��12�0 � 1 � sin�12�p�23

p�12�

1 0 � cos�12�p�23

p�12�

sin�12�p�23

p�12� cos�12�p�23

p�12� 0

�� 12 � g��12,�13�. (29)

The eigenvalue � i�� (�� 12[1 � (p�23/

p�12)2]12) and the corresponding eigenvector �i��

� � �12 � Gd g��12,�13�G] with the following threeelements,

G11d � � cos�12�p�23

p�12� � �1 � �p�23

p�12�2�

12,

G12d � sin�12�p�23

p�12� � �1 � �p�23

p�12�2�

12,

|τ12||τ23|

-4 -3 -2 -1 0 1 2 3 4x -3 -2 -1 0 1 2 3

y

0

4

8

12

-4 -2 0 2 4x-4-2 0 2 4

y

0.0

0.4

|τ13| 3π/2

ππ/2

0

-π/22ππ0

θ

θ12/θ23

|τ12||τ23|

-4 -3 -2 -1 0 1 2 3 4x -3-2

-1 0

1 2

3

y

0

4

8

-4 -2 0 2 4x -4 -2 0 2 4y

0.0

0.4

|τ13|

|τ12||τ23|

-4 -3 -2 -1 0 1 2 3 4x -3-2

-1 0

1 2

3

y

0

4

8

12

-4 -2 0 2 4x

-4 -2 0 2 4y

0.0

0.4

|τ13|

(a)

(b)

(c)

FIGURE 1. Profile of the non-adiabatic coupling ele-ments (�12� and �23�) with the separation of (a) 4 Å, (b) 2 Åand (c) 0.5 Å between (1, 2) and (2, 3) CI considering adi-abatic-to-diabatic transformation angles �12(x, y) � �23(x,y) � 1/2 tan�1(y/x)�1/4 sin[2 tan�1(y/x)] and �13(x, y) � 0.The inset in left of the figures show the non-adiabaticcoupling element �13�, while the inset on right of Figure1(a) represents the adiabatic-to-diabatic transformationangles. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]



G13d � � �1 � �p�23

p�12� 2�

12,

are substituted in the Eq. (25) to obtain the explicitform of rigorous EBO equation for the ground state.The functional form of the eigenvalue (��) of the

NAC matrix arising due to the above set of ADTangles are presented in Figures 2(b) and (c) for the4 and 0.5 Å separation of CIs, respectively with thefollowing analytic form:

�� x,y, x0� � � �12 1 �y4

4�4x02 � y2�2�1

� cos�2 tan�1y

2x0�2�

12,

and its’ gauge invariant conditions:

12��

0

2�

�� n,2x0 � 4�.d�n � 0.5

12��

0

2�

�� n,2x0 � 0.5�.d�n � 0.7

NUMERICAL CALCULATIONS: RESULTS ANDDISCUSSIONS

We consider the product between the groundvibrational state for the harmonic mode (at theasymptote of the scattering mode) and the Gaussianwavepacket with various KE energies for the scat-tering mode as the initial wavefunction for theground adiabatic state of the system. This adiabaticwavefunction is being propagated by using singlesurface BO approximate, approximate EBO, andrigorous EBO equations as functions of time withthe help of numerically accurate TDDVR [36] ap-proach and the respective wavefunction at t3 � isprojected on the asymptotic eigenfunctions of theHamiltonian to obtain the state-to-state vibrationaltransition probabilities at different energies. Weperform all those dynamical calculations at totalenergies 1.25, 1.50, and 1.75 eV. It is important tonote that all those equations (single surface BOapproximate, approximate EBO, and rigorous EBOequation) are derived with the assumption, namely,upper electronic states are expected to be classicallyclosed at those energies (1.25, 1.50, and 1.75 eV)with respect to the point of first CI at 3.0 eV.

On the other hand, we obtain the initial diabaticwavefunctions by performing the adiabatic–dia-batic transformation [Eq. (8)] on the wavefunctionmatrix, where the first and the second elements ofthis column matrix are (a) the adiabatic wavefunc-tion for the ground state and (b) zero, respectivelyat t � 0. Table I presents the reactive state-to-state

4 2 0 2 4x 1 0.5

0 0.5

1

y

0

5

10

15

u (eV)

-4 -3 -2 -1 0 1 2 3 4x -2-1

0 1

2

y

0

2

4

6

8

|ω|

-4 -3 -2 -1 0 1 2 3 4x -2-1

0 1

2

y

0

2

4

6

8

10

|ω|

(a)

(b)

(c)

FIGURE 2. (a) The three adiabatic potential energy sur-faces for Model A. The functional form of the eigenvalues(��) of the NAC matrix arising due to the ADT angles�12(x, y) � �23(x, y) � 1/2 tan � 1(y/x) � 1/4 sin[2tan � 1(y/x)] and �13 (x, y) � 0 for (b) 4 Å and (c) 0.5 Åseparation of CIs. [Color figure can be viewed in the on-line issue, which is available at www.interscience.wiley.com.]

SARKAR AND ADHIKARI


transition probabilities for the above choice of ADTangles with the separation between the CIs 4 and0.5 Å, respectively. Calculated transition probabili-ties by using single surface approximate and rigor-ous EBO equations with 4 Å separation of CIs notonly follow the correct symmetry (even3 odd orodd3 even) but also achieve quantitative agree-ment in comparison with diabatic results at all en-ergies. On the other hand, when the CIs are 0.5 Åapart, there is a gradual increase on even3 eventransitions along with even3 odd transitions andsome quantitative disagreement with diabatic tran-sitions particularly at higher energies as predictedby the gauge invariant condition [23]. Indeed, it isclear from Table I that single surface BO equationneither can provide correct symmetric transitionsnor can calculate quantitatively accurate probabili-ties, whereas transition probabilities calculated byusing approximate and rigorous EBO equationshave quantitatively similar agreements and/or dis-agreements with exact results (diabatic one).

The Model B

THE NON-ADIABATIC COUPLING ELEMENTSAND ADIABATIC PESS

The intention to formulate rigorous EBO equa-tion [Eq. (25)] is to consider the effect of upper

electronic states on the lower, particularly, whenthree electronic states are strongly coupled at apoint. We wish to construct a model (Model B) withsuch kind of coupling to explore the validity ofrigorous EBO equation [Eq. (25)] with respect to thediabatic [Eq. (6)] results as well as to find out thenecessity of rigorous EBO equation [Eq. (25)] withrespect to the approximate EBO equation [Eq. (26)].The 5 � 5 � matrix (1 2A�, 2 2A�, 3 2A�, 4 2A�, and 52A�) of H3 system breaks up to 3 � 3 � matrixleading to a typical example of three-state BO sys-tem. On this system, ab-initio calculations [37] dem-onstrate that �13 is relatively large and predict thatthe strongly overlapping �12 and �23 intersectionsessentially develops non-negligible value of �13, andthereby, this CI among the three adiabatic states isunbreakable and coincides at a point of degeneracy.

We construct the Model B by considering: (a)ADT angles [�12(n), �23(n), and �13(n)] definedaround a point; (b) Adiabatic PESs with degeneracyat the same point. We choose the following two setson the functional form of ADT angles to formulaterigorous EBO equation and construct diabatic PESs:

�I� �12�x,y� � �13�x,y� � �23�x,y� �12 tan�1�y

x��

14 sin�2 tan�1�y

x�;

TABLE I ______________________________________________________________________________________________Reactive state-to-state transition probabilities when (1,2) and (2,3) conical intersections are 4 Å, and 0.5 Å(2x0) apart.

2x0

1.25 eV 1.50 eV 1.75 eV

0 3 0 0 3 1 0 3 2 0 3 3 0 3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 0 0 3 1 0 3 2 0 3 3 0 3 4

4 Å 0.0005a 0.0510 0.0006 0.0492 0.0016 0.0216 0.0340 0.1089 0.0048 0.0000 0.0179 0.0007 0.0884 0.00040.0063b 0.0502 0.0010 0.0301 0.0045 0.0210 0.0069 0.1423 0.0045 0.0009 0.0175 0.0003 0.1000 0.00230.0025c 0.0500 0.0061 0.0384 0.0021 0.0206 0.0030 0.1080 0.0204 0.0006 0.0174 0.0034 0.0885 0.00360.0390d 0.0000 0.0619 0.0000 0.0600 0.0000 0.0939 0.0000 0.0138 0.0430 0.0000 0.0190 0.0000 0.1691

0.5 Å 0.0104 0.0250 0.0010 0.0280 0.0000 0.0686 0.0057 0.0539 0.0019 0.0102 0.0696 0.0282 0.1148 0.01940.0002 0.0251 0.0042 0.0734 0.0017 0.0695 0.0036 0.0770 0.0056 0.0050 0.0698 0.0057 0.0190 0.00260.0012 0.0252 0.0084 0.0631 0.0028 0.0692 0.0121 0.0694 0.0098 0.0047 0.0698 0.0011 0.0181 0.02120.0390 0.0000 0.0619 0.0000 0.0600 0.0000 0.0939 0.0000 0.0138 0.0430 0.0000 0.0190 0.0000 0.1691

a Diabatic.b rigorous EBO.c approximate EBO.d BO Approx.The diabatic surfaces are constructed considering the ADT angles, �12(x, y) � 1/2 tan�1(y/x � x0) � 1/4 sin[2 tan�1(y/x � x0)],�13(x, y) � 0, and �23(x, y) � 1/2 tan�1(y/x0 � x) � 1/4 sin[2 tan�1(y/x0 � x)]. The corresponding rigorous EBOs [Eq. (25)] and theapproximate EBOs [Eq. (26)] are derived under the same condition. Transition probabilities, when calculations are performed on thesingle adiabatic surface with BO approximation [Eq. (27)] are also presented.



�II� �12�x,y� � � 2 sin2�12 tan�1�y

x�, �13�x,y� �

�14 sin� tan�1�y

x� and �23�x,y� � tan�1�yx�

�12 sin�2 tan�1�y

x�.

The functional form of set (I) angles have verysimilar spatial distribution with the profile of ab-initio calculated ADT angles for H3 system [37]whereas the set (II) angles were being used else-where [22] to investigate any three-state BO systemdegenerate at a point. We substitute the set (I) and(II) ADT angles in the Eq. (10) to see the spatialdistributions of non-adiabatic coupling elementsand present in Figures 3 and 4, respectively. Theadiabatic SE can thus be constracted [see Eq. (4) andAppendix A of Ref. [23]) with the above sets ofNAC terms along with the following chosen formof adiabatic PESs [22],

u1� x,y� �12��0 � �1� x��2y2 � A1 � f� x,y�

u2� x,y� �12��0

2y2 � �D1 � A1� � f� x,y� � D1

u3� x,y� �12��0

2y2 � �D2 � A1� � f� x,y� � D2

�1� x� � �1exp�� x�1�2

f� x,y� � exp� �x2 � y2

�2 (30)

where the point of CI is at x � 0 and y � 0 [see Fig.5(a)] and potential parameters are same as Model A.

THE DIABATIC AND RIGOROUS EBOEQUATION

In a similar manner, each set of ADT angles [set(I)–(II)] constructs the ADT matrix [Eq. (8)] leading tothe transformation from adiabatic [Eq. (4)] to diabatic[Eq. (6)] SE, where the generalized expressions foradiabatic and diabatic SE were presented in Appen-dix A of Ref. 23. It may be noted that the transforma-tion matrices, A (�12, �23, �13)s [see Eq. (8)] beingdeveloped by using the above choices of ADT angles

ensure uniquely defined diabatic potential matrix inconfiguration space, i.e., the topological matrix, D �exp(�0

2� � � d�n), derived from ADT condition, shouldbe the unit one. The explicit expression of D [38] isformulated by using the G matrix as,

-0.4 -0.2 0 0.2 0.4x -0.4-0.2

0 0.2

0.4

y

0

20

40

60

80|τ12|

3π/2

π

π/2

0

-π/2 2ππ0θ

θ12

/θ23

/θ13

-0.4 -0.2 0 0.2 0.4x -0.4-0.2

0 0.2

0.4

y

0

10

20

30

|τ23|

-0.4 -0.2 0 0.2 0.4x -0.4-0.2

0 0.2

0.4

y

0

10

20

30

40

|τ13|

(a)

(b)

(c)

FIGURE 3. Profile of the non-adiabatic coupling ele-ments, (a) �12�, (b) �23�, and (c) �13� for the adiabatic-to-diabatic transformation angles �12(x, y) � �13(x, y) ��23(x, y) � 1/2 tan�1(y/x)�1/4 sin[2 tan�1(y/x)] when theconical intersections are situated at a point. The inseton right of Figure 3(a) represents the correspondingadiabatic-to-diabatic transformation angles. [Color fig-ure can be viewed in the online issue, which is avail-able at www.interscience.wiley.com.]

SARKAR AND ADHIKARI


D � �k�0

M

G�nk�E�nk�G†�nk�, (31)

where {nk, k � 0, 1, · · ·, M} is a series of pointsalong the contour (the last point nk is identical with

n0), G(nk) is the transformation matrix that diago-nalizes g(nk) and E(nk) is a diagonal matrix. Theform of the matrix, G†�nk� is shown in Eq. (22) andthe diagonal elements of E(nk) are given by, Ej(nk) �

exp��i�nk�1

nk �� j�n� � d�n�. While performing numericalcalculations to evaluate D matrix by using the two

-0.4 -0.2 0 0.2 0.4x -0.4-0.2

0 0.2

0.4

y

0

20

40

60

|τ12|2π

π

0

-π2ππ0

θ

θ12θ23θ13

-0.4 -0.2 0 0.2 0.4x -0.4-0.2

0 0.2

0.4

y

0

20

40

60

80

|τ23|

-0.4 -0.2 0 0.2 0.4x -0.4-0.2

0 0.2

0.4

y

0

20

40

60

80

100|τ13|

(a)

(b)

(c)

FIGURE 4. Profile of the non-adiabatic coupling ele-ments, (a) �12�, (b) �23�, and (c) �13� for the adiabatic-to-diabatic transformation angles �12(x, y) � �2 sin2[1/2 tan�1(y/x)], �13(x, y) � �1/4 sin[ tan�1(y/x)], �23 (x, y) �tan�1(y/x)�1/2 sin[2 tan�1(y/x)] when the conical intersec-tions are situated at a point. The inset on right of Figure4(a) represents the corresponding adiabatic-to-diabatictransformation angles. [Color figure can be viewed in theonline issue, which is available at www.interscience.wiley.com.]

2 1 0 1 2x 1 0.5

0 0.5

1

y

0

5

10

15

u (eV)

-3 -2 -1 0 1 2 3x -2-1

0 1

2

y

0

10

20

30

|ω|

-3 -2 -1 0 1 2 3x -2-1

0 1

2

y

0

5

10

15

20

25

|ω|

(a)

(b)

(c)

FIGURE 5. (a) The three adiabatic potential energy sur-faces for Model B. The functional form of the eigenvalues(��) of the NAC matrix arising due to the ADT angles(b) �12(x, y) � �23(x, y) � �13(x, y) � 1/2 tan � 1(y/x) � 1/4 sin[2 tan � 1(y/x)] (c) �12(x, y) � �2 sin2[1/2 tan � 1(y/x)],�13(x, y) � �1/4 sin[ tan � 1(y/x)], �23(x, y) � tan � 1(y/x) � 1/2 sin[2 tan � 1(yx)] when the conical intersections are situ-ated at a point. [Color figure can be viewed in the onlineissue, which is available at www.interscience.wiley.com.]



sets of ADT angles, it appears that in both cases, thematrix is unit one and the typical values of the Delements for the set (I) ADT angles are obtained as

D � � 1.000235 0.000000 0.0002020.000000 1.000322 0.000000

� 0.000498 0.000000 1.000127�.

Considering the CI of a three-states BO system ata point and introducing zero Curl ij around thesame point, the general form of non-adiabatic cou-pling matrix is given by Eq. (17) and thereby, theeigenvalues of the NAC matrices for the two sets((I)–(II)) of ADT angles appear as,

�� I � � �12�3 � 2 sin�12,

(32a)

�� II � � �12�1 �q � 4p2

p � 4�p sin��p�, q � �1

� �12�2, p � 1 � q, (32b)

where the functional form of the eigenvalues (��) ofthe NAC matrix arising from set (I) and (II) ADTangles are displayed in Figures 5(b) and (c), respec-tively. The above eigenvalue and the correspondingeigenvector with the following three elements,

G11d �

g3

�� cos�12�p�23

p�12� � sin�12cos�23�p�13

p�12�

� �1 � �p�13

p�12�2

� �p�23

p�12�2

� 2 sin�23�p�13

p�12��

12,

G12d � �

g2

�� sin�12�p�23

p�12� � cos�12cos�23�p�13

p�12�

� �1 � �p�13

p�12�2

� �p�23

p�12�2

� 2 sin�23�p�13

p�12��

12,

G13d �

g1

�� 1 � sin�23�p�13

p�12��1 � �p�13

p�12�2

� �p�23

p�12�2

� 2 sin�23�p�13

p�12��

12

are substituted in the Eq. (25) to obtain the explicitform of rigorous EBO equation for the ground state.On the other hand, the necessity of rigorous EBOequation [Eq. (25)] with respect to approximate

EBO equation [Eq. (26)] may be predicted from thefollowing gauge invariance,

�I �1

2��0

2�

�� I�n� � d�n � 1.030776,

�II �1

2��0

2�

�� II�n� � d�n � 1.090524. (33)

Since, for both the sets of ADT angles, the eig-envalues of the associated � matrix approximatelyobey the gauge invariance condition, namely unity,we expect both rigorous and approximate EBOequation can provide meaningful solutions but atthe same time, the rigorous EBO equation may notshow up much advantage over the approximateone. The general form of these gauge invarianceintegrals [Eq. (33)] along the integrands as definedin Eq. (32) are the so called incomplete elliptic in-tegral of the second kind and their nature is genericas long as the functional forms of the ADT anglesare analytic. This implies that when three electronicstates are coupled, the non-adiabatic effect of theupper states on the ground is equivalent to a po-tential developed due to an elliptic motion of thenuclei around the point of CI.

NUMERICAL CALCULATIONS: RESULTS ANDDISCUSSIONS

We initialize the ground adiabatic wavefunctionwith different initial KE as described in case ofModel A (see section Numerical Calculations: Re-sults and Discussions), propagate the wavefunctionand then, project the function at t3 � on the as-ymptotic eigenfunctions of the Hamiltonian to cal-culate state-to-state vibrational transition probabil-ities. Since the point of CI among the three-states isat 3.0 eV, it is expected that the upper electronicstates are classically closed with respect to theground at total energies 1.25, 1.50, and 1.75 eV andthereby, the formulated rigorous EBO equationsshould provide transition probabilities withenough accuracy. Tables II and III, present the re-active state-to-state transition probabilities for thetwo sets of ADT angles, respectively, where the firstrow displays the diabatic results, the second andthird rows show the transition probabilities calcu-lated by rigorous and approximate EBO equation,

SARKAR AND ADHIKARI


TABLE II _____________________________________________________________________________________________Reactive state-to-state transition probabilities when the conical intersection is situated at a point.

E(eV) 0 3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 5

1.25 0.0123a 0.0000 0.0782 0.00000.0125b 0.0089 0.0608 0.00730.0123c 0.0001 0.0745 0.05490.0390d 0.0000 0.0619 0.0000

1.50 0.0255 0.0000 0.0668 0.0000 0.0677 0.00000.0269 0.0012 0.0438 0.0026 0.0827 0.00420.0261 0.0075 0.0651 0.0118 0.0386 0.00290.0600 0.0000 0.0939 0.0000 0.0138 0.0000

1.75 0.0509 0.0000 0.0270 0.0000 0.0433 0.00000.0498 0.0073 0.0284 0.0085 0.0266 0.00810.0509 0.0077 0.0304 0.0315 0.0023 0.01890.0430 0.0000 0.0190 0.0000 0.1691 0.0000

a Diabatic.b rigorous EBO.c approximate EBO.d BO Approx.Diabatic surfaces are constructed considering the ADT angles, �12(x, y) � �13(x,y) � �23(x, y) � 1/2 tan�1 �y/x� � 1/4 sin�2 tan�1�y/x��. The corresponding rigorous EBOs [Eq. (25)] and the approximate EBOs [Eq. (26)] are derived under the same choiceof ADT angles. Transition probabilities, when calculations are performed on the single adiabatic surface with BO approximation [Eq.(27)] are also presented.

TABLE III ____________________________________________________________________________________________Reactive state-to-state transition probabilities when the conical intersection is situated at a point.

E(eV) 0 3 0 0 3 1 0 3 2 0 3 3 0 3 4 0 3 5

1.25 0.0275a 0.0000 0.0156 0.00000.0272b 0.0050 0.0158 0.00700.0275c 0.0001 0.0144 0.00420.0390d 0.0000 0.0619 0.0000

1.50 0.0371 0.0000 0.0077 0.0000 0.0356 0.00000.0372 0.0012 0.0200 0.0098 0.0177 0.01430.0371 0.0002 0.0065 0.0078 0.0255 0.00590.0600 0.0000 0.0939 0.0000 0.0138 0.0000

1.75 0.0237 0.0000 0.0208 0.0000 0.1400 0.00000.0244 0.0033 0.0101 0.0190 0.1099 0.01590.0238 0.0010 0.0178 0.0294 0.0884 0.05230.0430 0.0000 0.0190 0.0000 0.1691 0.0000

a Diabatic.b rigorous EBO.c approximate EBO.d BO Approx.Diabatic surfaces are constructed considering the ADT angles, �12(x, y) � �2 sin2 �1/2 tan � 1 �y/x��, �13 (x, y) � � 1/4 sin�tan � 1 �y/x��, �23 (x, y) � tan�1 �y/x� � 1/2 sin�2 tan � 1 �y/x��. The corresponding rigorous EBOs [Eq. (25)] and the approximateEBOs [Eq. (26)] are derived under the above choice of ADT angles. Transition probabilities, when calculations are performed on thesingle adiabatic surface with BO approximation [Eq. (27)] are also presented.



respectively with BO approximate results on thefourth row. As such, BO approximate results havevery little quantitative agreement with diabaticones. On the other hand, both rigorous and approx-imate EBO equations calculate accurate transitionprobabilities but those probabilities have similarlevel of agreement with respect to numerically ex-act diabatic results.

Summary

We have carried out BO treatment for N � 3 statesub-Hilbert space with single or multi-CI(s) BO sys-tem and formulated rigorous EBO equation for theground surface. Since we assume a coupled three-state electronic BO system constitutes a sub-Hilbertspace, it is possible to transform the adiabatic nu-clear SE to the diabatic one under the condition,� A � �A � 0, where the chosen form of thetransformation matrix (A) has to be orthogonal atany point in the configuration space and its’ ele-ments should be cyclic functions with respect to aparameter. Considering these natural requirementsto construct such A matrix, we take the product ofthree rotation matrices (where each matrix is con-stituted with Euler-like angle commonly calledADT angle) in six different way and substitute eachof these product matrices (As) in the ADT conditionto obtain six different sets of NAC elements interms of ADT angles. Each set of NAC elementssatisfy the Curl conditions with non-zero Diver-gences. It appears that a particular set of NACelements and their corresponding Curl-Divergenceequations can be reassigned to any other set withproper interchange of ADT angles. In our formula-tion, the connectivity among the adiabatic [Eq. (4)],the diabatic [Eq. (6)], and the rigorous EBO [Eqs.(25)] equation is through the ADT matrix [Eq. (8)].

The actual advantage of the explicit form of NACelements in terms of ADT angles lies in the deriva-tion of EBO equations. It appears that the formula-tion of rigorous EBO for any three-state BO systemis possible only when there exists coordinate inde-pendent ratio of the gradients for each pair of ADTangles. The validity of such ratio implies that theCurl of the vector field is zero. In this context, theexplicit form of the Curl equations in terms of ADTangles helps to explore analytically the validity ofzero Curls at and around the CIs. Defining theJacobian at and around the CIs, it has been possibleto show that the Curls are either identically or

approximately zero around the same region of nu-clear configuration space.

We have chosen various sets of ADT angles todefine the corresponding NAC terms and thereby,to mimic two situations of any three-state BO sys-tem, namely, (a) the point of CI between the firstand the second state is away from the CI betweenthe second and third (Model A) and (b) the point ofCI among the three-state is at a particular nuclearconfiguration (Model B). For both the models withdifferent choices of ADT angles, we have formu-lated the rigorous EBO equations by imposing zeroCurls (apart from Curl conditions being satisfied).The validity as well as necessity of rigorous EBOequation with respect to the approximate ones forboth the Models has been explored by performingnumerical calculations. Transition probabilities cal-culated by using diabatic and rigorous EBO equa-tions have good agreement among each other but assuch rigorous EBO results could not show up clearadvantage over the approximate one within thepresent model calculations. The rigorous approachessentially implies the formulation of Eq. (25) fromEq. (16) in a detailed manner, whereas in the pre-vious formulation [J Chem Phys 2006, 124, 074101],we write Eq. (26) in a intuitive sense directly fromEq. (16). Second, since the NAC terms could berapidly varying functions of nuclear coordinates, itis quite obvious that the nuclear coordinate depen-dence of G matrix can not be neglected in generalcases. For the specific examples studied in thepresent article, it appears that the elements of Gmatrix in the EBO equation do not show up prom-inent contribution in the transition probabilities butone can not predict it prior to any calculationsbased on Eq. (25). The major issue here the stepwiseformulation of eq. (16) to (25) and it’s numericalimplimentation, which is first kind of its nature.

ACKNOWLEDGMENTS

We acknowledge financial support from thethrough the project no. BS like to acknowledgesupport from program.

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