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**Higher order (A+E)**

**⊗e pseudo-Jahn–Teller coupling**

**⊗e pseudo-Jahn–Teller coupling**

### Wolfgang Eisfeld, Alexandra Viel

**To cite this version:**

### Wolfgang Eisfeld, Alexandra Viel. Higher order (A+E)

_{⊗e pseudo-Jahn–Teller coupling. Journal of}

_{⊗e pseudo-Jahn–Teller coupling. Journal of}

### Chemical Physics, American Institute of Physics, 2005, 122 (20), pp.204317. �10.1063/1.1904594�.

### �hal-01118370�

**Higher order**

### „

**A**

**+**

**E**

**…‹**

**e**

**pseudo-Jahn–Teller coupling**

Wolfgang Eisfelda兲

*Lehrstuhl für Theoretische Chemie, Department Chemie, Technische Universität München, D-85747*
*Garching, Germany*

Alexandra Vielb兲

*LPQ-IRSAMC, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse, France*

共Received 3 December 2004; accepted 17 March 2005; published online 25 May 2005兲

The pseudo-Jahn–Teller*共PJT兲 coupling of a nondegenerate state A with a twofold degenerate state*

*E by a degenerate vibrational mode e is studied for a general system with a C*3main rotational axis.

The PJT coupling terms up to sixth order are derived by symmetry considerations for this general

*共A+E兲*丢*e case. The obtained expression for the 3*⫻3 diabatic potential energy matrix is found to

be closely related to the expression recently developed for the higher order Jahn–Teller case关A. Viel
**and W. Eisfeld, J. Chem. Phys. 120, 4603**共2004兲兴. The dynamical PJT coupling, which can arise for
states of appropriate symmetry if one of the vibrational modes induces a change of the nuclear point
*group between D3h, C*3*v, C3h, and C*3, is discussed. The effect of the higher order PJT coupling is

*tested by a two-dimensional model study based on the e bending mode of NH*_{3}+. The models are
analyzed by fitting the two-dimensional potential energy surfaces. The significance of the higher
order terms on the nonadiabatic dynamics is demonstrated by quantum wave packet
*propagations. © 2005 American Institute of Physics.*关DOI: 10.1063/1.1904594兴

**I. INTRODUCTION**

Vibronic coupling, i.e., the interaction of energetically close-lying electronic states, induced by the vibrational mo-tion of the nuclei, often plays a very important role in spec-troscopy and nuclear dynamics. This is particularly true for radicals, which are usually characterized by a high density of electronic states, and molecules with high symmetry, result-ing in multiply degenerate states. Such vibronic interactions can induce nonadiabatic transitions among electronic states and thus may have a strong influence on the time evolution of the system.1,2A prominent example of such an interaction is the well-known Jahn–Teller共JT兲 effect.3,4In this case, the symmetry-induced degeneracy of an electronic state is lifted by distortions along modes of the appropriate symmetry and thus a conical intersection is formed. A conical intersection is characterized by the breakdown of the Born–Oppenheimer 共BO兲 approximation because the adiabatic wave functions of the interacting states cease to be continuously differentiable functions with respect to the nuclear coordinates and the symmetry point constitutes a pole of the nonadiabatic cou-pling elements.5

However, even without the existence of a conical inter-section the BO approximation may be invalid if two states become sufficiently close in energy and the nonadiabatic coupling elements become large. Besides the countless cases of general avoided crossings, the so-called pseudo-Jahn– Teller effect 共PJT兲 is particularly interesting for analyzing such nonadiabatic interactions. The PJT interaction is

char-acterized by a degenerate mode, coupling a degenerate with a
nondegenerate electronic state.1,6–9Such a situation can arise
*for any molecule with a main rotational axis C _{n}*

*with n*艌3.

*In the present study, the case of n = 3 is investigated.*

For the theoretical treatment of the nonadiabatic
dynam-ics of PJT systems, it is beneficial to change to the diabatic
representation of the electronic states.10,11 Although it was
shown to be impossible to uniquely define strictly diabatic
states,12,13 in practice a large number of diabatization
schemes has been developed and successfully applied.14–19
Due to the high symmetry of the PJT problem, a
“diabatiza-tion by ansatz” is particularly efficient and will be used
throughout this study. The elements of the diabatic potential
matrix are expanded as simple Taylor series of properly
sym-metrized nuclear coordinates and group theory is used to
determine general relations of the expansion coefficients.
The diabatic and adiabatic potential matrices are related by
the unitary transformation that diagonalizes the diabatic
ma-trix. Thus, the parameters of the diabatic matrix can be
ob-tained by fitting its eigenvalues to the adiabatic energies
*which are available, e.g., from ab initio calculations.*

Such an approach, which usually includes only the first 共sometimes also the second兲 nonvanishing coupling terms, is widely used for the treatment of JT and PJT problems 共see, e.g., Ref. 1 for an overview兲. From the dynamical point of view, the limitation to second-order terms along with using harmonic oscillator basis sets for the nuclear motion has the advantage that all terms of the Hamilton matrix can be evalu-ated analytically. However, the disadvantage is that for sys-tems displaying strong anharmonicity this method will give poor results for properties determined by more extended re-gions of the potential energy surfaces. Zgierski and Pawlikowski already noticed that for systems with both JT

a兲_{Electronic mail: wolfgang.eisfeld@ch.tum.de}

b兲_{Present address: PALMS-SIMPA, UMR 6627 du CNRS, Université de}

Rennes 1, Campus de Beaulieu, F-35042 Rennes, France; Electronic mail: alexandra.viel@univ-rennes1.fr

and PJT couplings, the second-order JT coupling needs to be included to obtain reasonable results when the first-order PJT coupling is considered.8In our previous study on the higher order JT couplings, we have shown that higher order cou-plings can have a drastic effect on the time evolution of such a system.20This suggests that higher order PJT coupling may also make significant contributions, particularly if higher or-der JT coupling is included.

The aim of the present study is to derive the higher order PJT coupling terms as was previously undertaken for the JT case. The method to determine the form of the coupling el-ements is based on the crude adiabatic approximation by Longuet-Higgens21 and was already used in the previous work on the JT couplings.20An alternative approach, based on the same fundamental considerations, was recently out-lined by Köppel.22

**II. SYMMETRY CONSIDERATIONS**

In the following, the diabatic potential matrix for the

*共A+E兲*丢*e pseudo-Jahn–Teller system up to sixth order is*

*derived for a general molecule with a C*3 main rotational

axis. Some general symmetry properties of the
*correspond-ing point groups D _{3h}, C*

_{3}

_{v}, C_{3h}, and C_{3}with respect to JT and PJT coupling are outlined in the following.

Common to systems of all four point groups are nuclear
coordinates belonging to a doubly degenerate vibrational
*mode e and doubly degenerate electronic states E. The *
in-variance of the total Hamiltonian under all symmetry
opera-tions of the point group along with simple application of
group theory leads to selection rules for the possible
cou-pling patterns. Thus it can be shown that only the two
*com-ponents of an e mode can couple the two comcom-ponents of a*
*doubly degenerate E state*共JT coupling兲. It is also easily seen
*that motion along a set of e coordinates is necessary to*
*couple a nondegenerate A state with an E state* 共PJT
cou-pling兲.

*While for the C*3group these are the only rules共because

*there is no distinction between different A and E *
representa-tions*兲 for C*3*v* *and D3h*further considerations apply. The six

*irreducible representations of the D3hgroup are A*1

### ⬘

*, A*2

### ⬘

*, E*

### ⬘

,*A*_{1}

### ⬙

*, A*

_{2}

### ⬙

*, and E*

### ⬙

*. It follows that the components of an E*

### ⬘

共or*E*

### ⬙

*兲 state can be coupled by an e*

### ⬘

mode in any order but only*by even orders of an e*

### ⬙

set of coordinates. For PJT coupling*a similar pattern is obtained. A*

_{1}

### ⬘

*and A*

_{2}

### ⬘

states can interact*with an E*

### ⬘

*state by any order in an e*

### ⬘

coordinate and by even*orders in an e*

### ⬙

mode. The same is true for interactions*be-tween A*

_{1}

### ⬙

*and A*

_{2}

### ⬙

*states with an E*

### ⬙

state. In contrast, for*coupling of A*

_{1}

### ⬘

*and A*

_{2}

### ⬘

*states with an E*

### ⬙

state coupling can*only occur by odd orders in an e*

### ⬙

mode. This is also true for*an interaction between A*

_{1}

### ⬙

*and A*

_{2}

### ⬙

*states with an E*

### ⬘

state.The rules given above have to be modified when more
than one vibrational mode is taken into account. A
*particu-larly frequent example is when an a*_{1}

### ⬙

*or a*

_{2}

### ⬙

coordinate is physically relevant. In this case, the PJT coupling between*pairs of states A*

_{共1,2兲}

### ⬘

*/ E*

### ⬙

*or A*

_{共1,2兲}

### ⬙

*/ E*

### ⬘

, respectively, will*strongly depend on the a*

_{共1,2兲}

### ⬙

coordinate共dynamical PJT cou-pling兲. Such a situation is analyzed in more detail in Sec. III B 3.*The degeneracy of the E state means that there exist two*
linearly independent eigenvectors 兩⌿_{1}典 and 兩⌿_{2}典 associated
to the same eigenvalue of the electronic Hamiltonian. Any
arbitrary, normalized linear combination of these two vectors
will again be an eigenvector to the same eigenvalue. This
*fact will be utilized in the following. However, in the D3h*

*and C*3*v*cases, there exist two particular linear combinations

兩⌿*x*典 and 兩⌿*y*典, which transform like different irreducible

*rep-resentations of the subgroups C*2*v共D3h兲 or Cs* *共C*3*v*兲.

**III.**„**A + E…‹e PSEUDO-JAHN–TELLER EFFECT**

**A. Complex representation**

The invariance of the total Hamiltonian under the
sym-metry operations of the symsym-metry point group of the system
is used to obtain the nonvanishing terms of the potential
energy matrix. The complex representation for both the
vi-brational degrees of freedom and the electronic states
*simpli-fies the study of the effect of the Cˆ*3symmetry operator. A set

*of complex coordinates, corresponding to the e vibrational*
*mode, is obtained from the real coordinates x and y by the*
transformation,

*Q*_{+}*= x + iy ,* 共1a兲

*Q*−*= x − iy .* 共1b兲

The two orthogonal components of the electronically
*degen-erate E state*共具⌿1兩 and 具⌿2兩兲 are modified by the

correspond-ing unitary transformation whereas the nondegenerate
elec-tronic eigenfunction共具⌿*a*兩兲 remains unaffected,

**U**†⌿* _{共a12兲}*=

_{冑}

1
2### 冢

### 冑

2 0 0 0 1*i*0

*1 − i*

### 冣

### 冢

具⌿*a*兩 具⌿1兩 具⌿2兩

### 冣

=### 冢

具⌿*a*兩 具⌿+兩 具⌿−兩

### 冣

=⌿*. 共2兲*

_{共a+−兲}*In this complex representation both the coordinates Q*+

*, Q*−

and the state functions 具⌿+兩, 具⌿−兩 are eigenfunctions of the

*symmetry operator Cˆ*_{3} *with eigenvalues e±2i*/3_{. In contrast,}

the具⌿*a*兩 state function is invariant with respect to this

opera-tion. Thus, a rotation of 2/ 3 transforms the complex coor-dinates as

*Cˆ*3*Q*+*= e*+2*i/3Q*+, *Cˆ*3*Q*−*= e*−2*i/3Q*−, 共3兲

and the electronic state functions as

*Cˆ*_{3}具⌿* _{a}*兩 = 具⌿

_{a}兩, Cˆ_{3}兩⌿

*典 = 兩⌿*

_{a}*典, 共4a兲*

_{a}*Cˆ*3具⌿+*兩 = e*−2*i/3*具⌿+*兩, Cˆ*3兩⌿+*典 = e*+2*i/3*兩⌿+典, 共4b兲

*Cˆ*3具⌿−*兩 = e*+2*i/3*具⌿−*兩, Cˆ*3兩⌿−*典 = e*−2*i/3*兩⌿−典. 共4c兲

The representation of the electronic Hamiltonian in the

兵兩⌿*a*典,兩⌿+典,兩⌿−典其 basis is given by

*Hˆ*el=⌿* _{共a+−兲}*
†

**H**

*⌿*

_{共a+−兲}*=*

_{共a+−兲}### 兺

*i,j*兩⌿

*i典Hij*具⌿

*j兩 共i, j = a, + ,− 兲,*共5兲

*and the matrix elements H*=具⌿

_{ij}

_{i}兩Hˆ_{el}兩⌿

_{j}典 共i, j=a, + ,−兲 are*expanded as Taylor series up to sixth order in Q*+

*and Q*−.

Each term of the expansion for each element of Eq.共5兲 has to fulfill the invariance condition when applying the 2/ 3 ro-tation operator. For example,

*Cˆ*_{3}兩⌿_{a}典Q_{+}*pQq*_{−}具⌿_{+}*兩 = 1 ⫻ e共+p兲2i/3e共−q兲2i/3e*−2*i/3*

⫻兩⌿*a典Q*+

*p _{Q}*
−

*q*

_{具⌿}

+兩 共6兲

can contribute only for*共p,q兲 combinations which fulfill the*
condition *共p−q−1兲共mod 3兲=0. The nonvanishing terms of*
*the matrix Hij*are given in Table I in which the Jahn–Teller

terms obtained in Ref. 20 have been reported again for
com-pleteness. Note that the nonvanishing terms of the matrix
*element Ha+are the same as the ones of H*+−.

23

Next, the invariance of the nonvanishing terms with re-spect to the other symmetry operations of the symmetry group needs to be tested. To this end, the obtained Hamil-tonian matrix is transformed back to the real representation.

**B. Real representation**

The real representation for both the electronic state func-tions and the nuclear coordinates are often preferred for the

*ab initio computation of electronic energies and for the *

dy-namics studies. The matrix representation*H of the electronic*
Hamiltonian in the real representation, including all Jahn–
Teller and pseudo-Jahn–Teller couplings up to sixth order, is
obtained by the back transformation,

*Hˆ*el=⌿* _{共a+−兲}*
†

**U**†**UH**_{共a+−兲}**U**†**U**⌿* _{共a+−兲}*=⌿

*†*

_{共a12兲}*H⌿*. 共7兲 This results in the factorized expression:

_{共a12兲}* H = UHU*†

_{=}

### 兺

*n=0*6 1

*n!*

## 冦冢

*VA共n兲*0 0 0

*V*0 0 0

_{E}共n兲*VE共n兲*

## 冣

+### 冢

0 0 0 0*W*JT

*共n兲*

*Z*JT

*共n兲*0

*Z*

_{JT}

*共n兲*−

*W*

_{JT}

*共n兲*

### 冣

+## 冢

0*W*

_{PJT}

*共n兲*−

*Z*

_{PJT}

*共n兲*

*W*PJT

*共n兲*0 0 −

*Z*

_{PJT}

*共n兲*0 0

## 冣冧

=### 兺

*n=0*6 1

*n!*

**兵V**diag

*共n兲*

**JT**

_{+ V}*共n兲*

**PJT**

_{+ V}*共n兲*

_{其.}

_{共8兲}

**The diagonal matrices V**_{diag}*共n兲* represent the potentials of the
corresponding states in absence of any coupling. The
**matri-ces V**_{JT}*共n兲* are responsible for the splitting of the degenerate
**state due to the Jahn–Teller effect whereas V**_{PJT}*共n兲* are the
pseudo-Jahn–Teller coupling matrices. The explicit
param-etrized expressions for the diagonal *V共n兲* elements and the
coupling elements*W共n兲*and*Z共n兲*are given by

*V*共0兲* _{= a}*
1
共0兲

_{,}

_{共9a兲}

*V*共1兲

_{= 0,}

_{共9b兲}

*V*共2兲

*1 共2兲*

_{= a}*2*

_{关x}*2*

_{+ y}_{兴,}

_{共9c兲}

*V*共3兲

*1 共3兲*

_{= a}*3*

_{关2x}*2*

_{− 6xy}_{兴,}

_{共9d兲}

*V*共4兲

*1 共4兲*

_{= a}*4*

_{关x}*2*

_{+ 2x}*2*

_{y}*4*

_{+ y}_{兴,}

_{共9e兲}

*V*共5兲

*1 共5兲*

_{= a}*5*

_{关2x}*3*

_{− 4x}*2*

_{y}*4*

_{− 6xy}_{兴,}

_{共9f兲}

*V*共6兲

*1 共6兲*

_{= a}*6*

_{关2x}*4*

_{− 30x}*2*

_{y}*2*

_{+ 30x}*4*

_{y}*6*

_{− 2y}_{兴}

*+ a*

_{2}共6兲

*关x*6

*+ 3x*4

*y*2

*+ 3x*2

*y*4

*+ y*6兴, 共9g兲

*W*共0兲

_{= 0,}

_{共10a兲}

*W*共1兲

_{=}

_{}1 共1兲

_{x,}_{共10b兲}

*W*共2兲

_{=}

_{}1 共2兲

*2*

_{关x}*2*

_{− y}_{兴,}

_{共10c兲}

*W*共3兲

_{=}

_{}1 共3兲

*3*

_{关x}*2*

_{+ xy}_{兴,}

_{共10d兲}

*W*共4兲

_{=}

_{}1 共4兲

*4*

_{关x}*2*

_{− 6x}*2*

_{y}*4*

_{+ y}_{兴 + }2 共4兲

*4*

_{关x}*4*

_{− y}_{兴,}

_{共10e兲}

*W*共5兲

_{=}

_{}1 共5兲

*5*

_{关x}*3*

_{− 10x}*2*

_{y}*4*

_{+ 5xy}_{兴 + }2 共5兲

*5*

_{关x}*3*

_{+ 2x}*2*

_{y}*4*

_{+ xy}_{兴,}共10f兲

*W*共6兲

_{=}

_{}1 共6兲

*6*

_{关x}*4*

_{− 5x}*2*

_{y}*2*

_{− 5x}*4*

_{y}*6*

_{+ y}_{兴}+

_{2}共6兲

*关x*6

*+ x*4

*y*2

*− x*2

*y*4

*− y*6兴, 共10g兲

*Z*共0兲

_{= 0,}

_{共11a兲}

*Z*共1兲

_{=}

_{}1 共1兲

_{y ,}_{共11b兲}

*Z*共2兲

_{= − 2}

_{}1 共2兲

_{xy ,}_{共11c兲}

TABLE I. Nonvanishing terms of the 3⫻3 Hamiltonian matrix in complex
representation.
Order *Diagonal Haa= H*++*= H*−−
*Off-diagonal Ha−= H+a= H*−+
=*共Ha+*兲*=*共H−a*兲*=*共H*+−兲*
0 *Q*+
0
*Q*−
0
—
1 — *Q*+
0* _{Q}*
−
1
2

*Q*+ 1

*Q*− 1

*Q*+ 2

*Q*− 0 3

*Q*+ 3

*− 0*

_{Q}*+ 0*

_{and Q}*− 3*

_{Q}*+ 1*

_{Q}*− 2 4*

_{Q}*Q*+ 2

*Q*− 2

*Q*+ 0

*Q*− 4

*and Q*+ 3

*Q*− 1 5

*Q*+ 4

*− 1*

_{Q}*+ 1*

_{and Q}*− 4*

_{Q}*+ 2*

_{Q}*− 3*

_{Q}*+ 5*

_{and Q}*− 0 6*

_{Q}*Q*+ 6

*Q*− 0

*and Q*+ 3

*Q*− 3

*and Q*+ 0

*Q*− 6

*Q*+ 1

*Q*− 5

*and Q*+ 4

*Q*− 2

*Z*共3兲_{=}_{}
1
共3兲* _{关x}*2

*3*

_{y + y}_{兴,}

_{共11d兲}

*Z*共4兲

_{=}

_{}1 共4兲

*3*

_{关4x}*3*

_{y − 4xy}_{兴 + }2 共4兲

*3*

_{关− 2x}*3*

_{y − 2xy}_{兴,}

_{共11e兲}

*Z*共5兲

_{=}

_{}1 共5兲

*4*

_{关− 5x}*2*

_{y + 10x}*3*

_{y}*5*

_{− y}_{兴 + }2 共5兲

*4*

_{关x}*2*

_{y + 2x}*3*

_{y}*5*

_{+ y}_{兴,}共11f兲

*Z*共6兲

_{=}

_{}1 共6兲

*5*

_{关4x}*5*

_{y − 4xy}_{兴 + }2 共6兲

*5*

_{关− 2x}*3*

_{y − 4x}*3*

_{y}*5*

_{− 2xy}_{兴.}共11g兲 It is worth noting that the elements of the JT and PJT cou-pling matrices are of identical form and are only distin-guished by different coupling constants

*. It now needs to be tested whether the Hamiltonian in the real representation fulfills the invariance condition with respect to the remaining symmetry operations of the nuclear point group.*

_{m}共n兲**1. D3hSymmetry**

*The most restrictive rules apply in the case of D3h*

*sym-metry. Besides the Cˆ*_{3}rotation, this group is characterized by
*the operators Cˆ*2,*ˆv*, *ˆh, and Sˆ*3*. The Sˆ*3 operator is special

*insofar that it corresponds to a successive application of Cˆ*3

and*ˆ _{h}*. However, the Hamiltonian derived in the preceding

*section is constructed to be invariant under the Cˆ*3operation

*and thus the symmetry properties with respect to Sˆ*3are fully

equivalent to those with respect to*ˆ _{h}. Similarly, the Cˆ*

_{2}ro-tation can also be achieved by successive application of

*ˆ*

_{v}and*ˆ _{h}* and thus it is sufficient to test the invariance of the
Hamiltonian under the latter two operations. The symmetry

*properties of the x and y nuclear coordinates as well as the*ones resulting for the

*V, W, and Z functions are given in*Table II. The characters for the electronic state functions are also presented in Table II. Here the fact is used that the

*components E*

_{x}### ⬘

*and E*

_{y}### ⬘

*can be chosen to transform as A*1and

*B*2*in the subgroup C*2*vwhile Ex*

### ⬙

*and Ey*

### ⬙

*transform as B*1and

*A*2. The same correspondence is used for the degenerate

*co-ordinates e*

### ⬘

*and e*

### ⬙

.*For a general D3h* symmetric molecule there may exist

*coordinates of e*

### ⬘

*as well as e*

### ⬙

character, though in the most*common examples of AB*3

*molecules only e*

### ⬘

coordinates are*present. The e*

### ⬘

*and e*

### ⬙

sets of coordinates are distinguished by their transformation under*ˆ*

_{h}. It follows that for e### ⬘

coor-dinates all orders of*V, W, and Z are invariant under*

*ˆ*

_{h}*operation. In contrast, for an e*

### ⬙

mode only even orders of*V,*

*W, and Z are invariant while all odd orders are *

antisymmet-ric. For the*ˆ _{v}operator there is no distinction between the e*

### ⬘

*and e*

### ⬙

case but it is found that*V and W are invariant and Z*changes sign under this operation.

The total Hamiltonian is invariant if each term of the

form 兩⌿* _{i}典H_{ij}*典⌿

_{j}兩 with i, j苸兵a,1,2其 is totally symmetricwith respect to the operations given in Table II. Thus the
imposed symmetry of the matrix elements *H _{ij}* depends on
the symmetry of the electronic states involved in the
cou-pling. It is easily seen that the condition

*共Hij*兲 =共⌿*i*兲共⌿*j*兲 共12兲

must be fulfilled for the *ˆ _{h}* and

*ˆ*operations where the characterscan be taken from Table II.

_{v}A first trivial result is that all diagonal elements *Haa*,

*H*11, and*H*22 must be totally symmetric. The element*H*12,

for which the symmetry is given by ⌫_{⌿}

1丢⌫⌿2丢⌫*Z*, is

al-ways invariant for arbitrary assignments 兵⌿1,⌿2*其↔*

兵⌿*x*,⌿*y*其. The only limitation applies to coupling through an

*e*

### ⬙

mode in which case only even orders of*Z共n兲*are possible. For

*Ha1*and

*Ha2*the explicit form of the coupling elements

*depends on the symmetry of the A state and the assignment*

of兵⌿1,⌿2其. The PJT coupling elements for all possible

com-binations of states are given in Table III. Depending on the
symmetry of the coupling mode *共e*

### ⬘

*/ e*

### ⬙

兲, certain couplings vanish or are limited to even or odd orders of the coupling functions*W共n兲*and

*Z共n兲*which is also indicated in the table. Table III along with Eq.共8兲 enables us to construct di-abatic potential matrices for any combination of state sym-metries and coupling modes which fulfill the invariance con-dition of the total Hamiltonian. As an example, for coupling

*of an A*

_{1}

### ⬘

*with an E*

### ⬘

*state by an e*

### ⬘

mode it is found that the*E state components have to be assigned as* 兩⌿1典=兩⌿*x*典 and

兩⌿2典=兩⌿*y典. All orders n are allowed which result in 32 free*

coupling parameters for a full sixth-order coupling matrix. If
*the A*_{1}

### ⬘

*state is replaced by A*

_{2}

### ⬘

, only the assignment of兩⌿1典and 兩⌿2典 needs to be interchanged while the form of the

diabatic matrix remains the same.

TABLE II. Symmetry properties of the nuclear coordinates, of the nonzero
matrix elements*共n⬎0 for W and Z兲 and of the possible electronic state*
*functions in D3h*symmetry.
*Cˆ*_{2} *ˆv* *ˆh*
*x共ex*⬘*, ex*⬙兲 +1, −1 +1 +1, −1
*y共ey*⬘*, ey*⬙兲 −1, +1 −1 +1, −1
*V共n兲 _{共e}*

_{⬘}

_{兲}

_{+1}

_{+1}

_{+1}

*W共n兲*

_{共e}_{⬘}

_{兲}

_{+1}

_{+1}

_{+1}

*Z共n兲*

_{共e}_{⬘}

_{兲}

_{−1}

_{−1}

_{+1}

*V共n兲 _{共e}*

_{⬙}

_{兲}

_{+1}

_{共even n兲; −1 共odd n兲}_{+1}

_{+1}

_{共even n兲; −1 共odd n兲}*W共n兲*

_{共e}_{⬙}

_{兲}

_{+1}

_{共even n兲; −1 共odd n兲}_{+1}

_{+1}

_{共even n兲; −1 共odd n兲}*Z共n兲*

_{共e}_{⬙}

_{兲}

_{−1}

_{共even n兲; +1 共odd n兲}_{−1}

_{+1}

_{共even n兲; −1 共odd n兲}*Ex*⬘ +1 +1 +1
*Ey*⬘ −1 −1 +1
*A*1⬘ +1 +1 +1
*A*2⬘ −1 −1 +1
*Ex*⬙ −1 +1 −1
*Ey*⬙ +1 −1 −1
*A*1⬙ +1 −1 −1
*A*2⬙ −1 +1 −1

TABLE III. Nonvanishing coupling elements*Hij, i, j苸兵a,x,y其 in the real*

*representation for D3h*symmetry. Given in parentheses are the possible order

*of the coupling elements for coupling by e*⬘*/ e*⬙modes.

*Ex*⬘ *Ey*⬘ *Ex*⬙ *Ey*⬙

*A*1⬘ *W共n兲*共all/even兲 *Z共n兲*共all/even兲 *W共n兲*共—/odd兲 *Z共n兲*共—/odd兲

*A*2⬘ *Z共n兲*共all/even兲 *W共n兲*共all/even兲 *Z共n兲*共—/odd兲 *W共n兲*共—/odd兲

*A*1⬙ *Z共n兲*共—/odd兲 *W共n兲*共—/odd兲 *Z共n兲*共all/even兲 *W共n兲*共all/even兲

*A*2⬙ *W共n兲*共—/odd兲 *Z共n兲*共—/odd兲 *W共n兲*共all/even兲 *Z共n兲*共all/even兲

**2. C3v, C3h, and C3symmetry**

*C*3*v, C3h, and C*3 *are subgroups of D3h* for which the

diabatic matrices can be easily obtained from the derivation
*of the D3hcase. In C*3*v* the only symmetry operator besides
*Cˆ*3 is*ˆvand the resulting irreducible representations are A*1,

*A*2*, and E. The Ex* *and Ey* components can be chosen to

*transform like A*

### ⬘

*and A*

### ⬙

*of the subgroup Cs. Thus, the A*1

*/ E*

coupling is described by the same diabatic matrix as the

*A*_{1}

### ⬘

*/ E*

### ⬘

*pair of states coupled by an e*

### ⬘

*mode in D3h*. The

*coupling of A*2*/ E corresponds to the* *共A*2

### ⬘

*+ E*

### ⬘

兲丢*e*

### ⬘

case.*The C3h* group is characterized by the symmetry

*opera-tors Cˆ*3,*ˆh, and Sˆ*3 which yields the four irreducible

*repre-sentations A*

### ⬘

*, E*

### ⬘

*, A*

### ⬙

*, and E*

### ⬙

*. The distinction between x and*

*y components for the degenerate electronic states is useless*

because there is no operator available to decide about the
assignment. Thus the coupling patterns are the same as for
*the C*3*v* case, neglecting the distinctions related to the

sub-scripts 1 and 2.

*In C*3*, Cˆ*3is the only operator and there is no distinction

*between different A states. Here also no operator can *
*dis-criminate between the Ex* *and Ey* components. Again, this

*case is described by the same diabatic matrix as for C*3*v*

*symmetry, just without any reference to Exand Ey*and to the

subscripts 1 and 2.

**3. D3hsymmetry and dynamical pseudo-Jahn–Teller**

**effect**

As indicated in Table III, PJT coupling between states having different characters共

### ⬘

versus### ⬙

兲 with respect to the*ˆ*

_{h}*operator can only occur by displacement along an e*

### ⬙

mode. However, many systems of interest, for which JT and PJT*couplings are likely, are AB*3

*molecules for which no e*

### ⬙

*mode exists. For example the A*_{2}

### ⬙

electronic ground state of NH_{3}+

*is not coupled to the A˜ E*

### ⬘

*excited state by the e*

### ⬘

mode*since A*

_{2}

### ⬙

丢*E*

### ⬘

丢*E*

### ⬘

*does not contain A*

_{1}

### ⬘

. However, this picture is only valid if coupling by only a single vibrational mode is considered. In fact, these two states are coupled by*simulta-neous motions in the a*

_{2}

### ⬙

mode, which pyramidalizes the*mol-ecule, and an e*

### ⬘

*mode because A*

_{2}

### ⬙

丢*E*

### ⬘

丢*A*

_{2}

### ⬙

丢*E*

### ⬘

*傻A*

_{1}

### ⬘

. By*allowing nonplanar geometries, the effect of the a*

_{2}

### ⬙

bending can be seen as switching on the PJT coupling caused by*displacement along the degenerate e*

### ⬘

mode. Thus, this*situ-ation can be called a dynamical pseudo-Jahn–Teller effect.*The electronic Hamiltonian, introducing the coupling due to

*the umbrella coordinate Qa*

2

⬙at first order, is given by

*H =*

### 兺

*n=0*6 1

*n!*

**兵V**diag

*共n兲*

**JT**

_{+ V}*共n兲*

_{+ Q}*a*2 ⬙

**V**PJT

*共n兲*其 +

### 冢

*UA*0 0 0

*UE*0 0 0

*UE*

### 冣

. 共13兲*U are even functions of the umbrella coordinate only and*

*represent the unperturbed potentials of the A and E states*
*along the a*_{2}

### ⬙

mode. In a multidimensional study of NH_{3}+, which is a prototype example for dynamical PJT coupling, such a significant variation of the PJT coupling with the um-brella mode must be taken into account.

**IV. APPLICATION**

The derived diabatic matrix is applied to the
*representa-tion of a two-dimensional cut along the e bending mode of*
the two lowest electronic states of the NH_{3}+cation. These two
states correspond to 2*A*_{2}

### ⬙

symmetry for the ground state and2_{E}

_{⬘}

_{symmetry for the excited state. Although the ground}

state geometry of the cation is trigonal planar, the pyramidal

*C*3*v* ground state geometry of the neutral NH3 is chosen as

*reference geometry for the cuts. In C*3*v*the state symmetries

are 2*A*_{1} and2*E. This pyramidal structure is indeed relevant*

when treating, for example, the ionization process. We also
investigated cuts for different pyramidalization angles which
influences the strength of the PJT coupling by the discussed
*dynamical effect. Cuts along the other e vibrational*

共stretch-ing兲 mode have also been considered. The presented

two-dimensional cut has been found to be the most suitable to perform the following model study.

The reference data for the adiabatic potential energies
*along the two components of the e bending coordinate are*
obtained by multireference configuration interaction共MRCI兲
calculations, using the augmented cc-pVQZ basis set. Since
the structure is pyramidalized, the bending angles are defined
by the projection of the three N–H bonds onto the plane
which is normal to the trisector, i.e., the line that forms equal
angles with all three N–H bonds. Using these projected
*angles, the e coordinates are defined by displacements from*
*the C*3*v* *reference geometry as x = 6*−1/2共2⌬␣1−⌬␣2−⌬␣3兲

*and y = 2*−1/2共⌬␣2−⌬␣3兲. All calculations have been

per-formed by the MOLPROprogram.24

The calculated potential cuts show very strong
anharmo-nicity, so that a higher order expansion is necessary to
accu-rately describe the system. In order to decide to which order
the expansion has to be done, a stepwise scheme is chosen.
We first investigate the influence of the Jahn–Teller coupling
order on the description of the potentials, setting the
pseudo-Jahn–Teller coupling to zero. To this end, the parameters of
the diabatic matrix have to be optimized such that the error
*between the eigenvalues of the matrix and the ab initio *
en-ergies is minimized. This is achieved by nonlinear
least-squares fitting which is performed by a combination of a
standard Marquardt–Levenberg procedure embedded into a
genetic algorithm.

*In Fig. 1 the ab initio data are displayed together with*
the fitting results for the models JT2, JT4, and JT6 along the

*x coordinate. The notation JTNmax corresponds to *

*expan-sion up to n = Nmax for the diagonal* *V共n兲* and Jahn–Teller
coupling matrix elements *W _{JT}共n兲*,

*Z*in Eq.共8兲. In these cal-culations, all PJT couplings are set to zero. It is immediately evident that the second-order model 共JT2兲 gives very poor results for all three potentials, except for a very narrow re-gion around the reference geometry. The fourth-order ap-proximation 共JT4兲 improves the results drastically, though deviations with respect to the reference data are clearly ob-servable. Finally, the sixth-order expansion 共JT6兲 yields

_{JT}共n兲*po-tentials which are virtually indistinguishable from the ab*

*ini-tio data.*

Though such a good agreement in general is very desir-able, one has to keep in mind that in the real system

pseudo-Jahn–Teller coupling also contributes to the shape of the po-tentials. Thus, the question arises to which degree the anharmonicity has to be accounted for by JT or PJT cou-pling, respectively. For example, the almost perfect JT6 fit indicates that adding further correction terms by the PJT cou-plings could render the fitting function too flexible. Thus, the obtained parameters may loose part of their physical signifi-cance. In such a case, a careful analysis of the obtained po-tential matrix, e.g., by using properties of the electronic wave functions, can help to decide about the suitable coupling

or-ders共see below兲.

We tested all different combinations of JT and PJT cou-pling and depict characteristic results in Fig. 2. The lower panel of Fig. 2 shows a combination of JT4 and PJT1 which is comparable to the JT4/PJT2 result 共not depicted兲. The adiabatic energies 共indicated by dashed lines with open circles兲 still deviate somewhat from the reference data. Char-acteristic is that two diabatic energy curves 共indicated by

dotted lines with open diamonds兲 cross at a displacement of
roughly 1.2 a.u. The middle panel represents the results for
the JT4/PJT4 model. The agreement of the adiabatic energies
is significantly improved and deviates only very slightly for
*the Eystate component for negative x* *共the Ey* state

*compo-nent presents a minimum at negative x*兲. The JT4/PJT6
model共not depicted兲 shows no significant improvement over
the JT4/PJT4 combination. However, the JT6/PJT4 model
yields almost perfect adiabatic energies while the shape of
the diabatic potentials is not qualitatively different from the
JT4/PJT4 or JT4/PJT6 results. For these three models, no
*crossing of the diabatic surfaces at positive x is obtained. As*
a third example, the JT6/PJT6 model is presented in the
up-per panel of Fig. 2. The adiabatic energies are now in almost
perfect agreement with the reference data. The interesting
difference to the other models is that a crossing of diabatic
potentials is found at about −0.7 a.u.

It is also helpful to compare the coupling coefficients

*FIG. 1. Adiabatic energies in eV along x for the pure Jahn–Teller models of*
order 2共lower panel兲, 4 共middle panel兲, and 6 共upper panel兲 共dashed lines
with open circles*兲 compared to the ab initio reference data 共solid line兲.*

FIG. 2. Adiabatic共dashed lines with open circles兲 and diabatic 共dotted lines
with open diamonds*兲 energies in eV along x for JT4/PJT1 共lower panel兲,*
JT4/PJT4共middle pannel兲, and JT6/PJT6 共upper panel兲 models compared to
*the ab initio reference data*共solid line兲.

obtained for the different models. The values for the PJT and
JT couplings for some characteristic cases are collected in
Table IV共all parameters are given in atomic units兲. The
lin-ear coupling constant_{1}共1兲is present in all models and Table
IV shows that for all models except JT6/PJT4 and JT6/PJT6
the values are rather similar. It also becomes apparent that
the coefficients for the higher couplings are all of significant
magnitude. However, one should keep in mind that the
fac-torial factor in Eq. 共8兲 reduces the actual coupling strength
for the higher order couplings. Comparison of the higher
order parameters shows that the values for different models
are in the same order of magnitude but in some cases change
sign. Particularly the JT4/PJT6 and JT6/PJT6 values show a
significantly different behavior. It is likely that in the JT6/
PJT6 case the model is already too flexible to yield
param-eters of physical significance. This will be illuminated from a
different view point in the following.

The fitting error with respect to the adiabatic energies
itself is not a suitable measure because JT6/PJT4 and JT6/
PJT6 result in comparably small errors. However, the
diaba-tic states vary strongly with the order of the JT and PJT
model and thus a criterion is needed to decide which model
is relevant. One can obtain useful information from the
*elec-tronic wave functions of the ab initio calculations. The CI*
vectors of complete active space self-consistent field
共CASSCF兲 or MRCI calculations clearly show whether or
not the character of two electronic states is interchanged
upon distortion along a given coordinate. A
maximum-overlap criterion can be applied to the CASSCF molecular
orbitals with respect to a reference wave function in which
no coupling of the states in question can occur共e.g., due to
symmetry*兲. In the basis of these so-called diabatic orbitals,*
the CI coefficients of the treated states reflect their
coupling.16 The results of such a calculation for
*displace-ments along the x coordinate is depicted in Fig. 3.*

Since only the2*A*_{1}and2*E _{x}*state components are coupled

*along x, it is sufficient to monitor the CI coefficients of the*
corresponding two main electron configurations. The
result-ing 2⫻2 matrix is reorthonormalized and the obtained CI
*coefficients are plotted against the displacement in x. From*
this plot it can be seen that the2*A*_{1}and2*E _{x}*state components

*start to mix significantly for positive x displacements and to*

*a lesser extent along negative x distortions. However, most*important is the finding that apparently no intersection of the configurations is observed. Thus, the two state components keep their main character throughout the whole range of the displacement without a crossing of the diabatic states as well. On the grounds of this analysis it is seen that the best

TABLE IV. PJT and JT coupling coefficients共a.u.兲 determined for different models.

JT2/PJT1 JT4/PJT1 JT4/PJT4 JT6/PJT4 JT4/PJT6 JT6/PJT6
1
共1兲_{共PJT兲} _{0.0441} _{0.0548} _{0.0549} _{0.1061} _{0.0467} _{0.1648}
1
共2兲_{共PJT兲} _{−0.2037} _{−0.2689} _{−0.2926} _{−0.3989}
1
共3兲_{共PJT兲} _{−0.2756} _{0.1521} _{0.8040} _{−0.2183}
1
共4兲_{共PJT兲} _{−0.1482} _{0.1925} _{0.2888} _{0.4670}
2共4兲共PJT兲 1.3420 0.3858 1.6040 3.3783
1
共5兲_{共PJT兲} _{−1.1990} _{−8.2329}
2
共5兲_{共PJT兲} _{−9.9024} _{1.7840}
1
共6兲_{共PJT兲} _{7.6403} _{2.4826}
2
共6兲_{共PJT兲} _{2.4144} _{−3.7752}
1共1兲共JT兲 0.1485 0.1293 0.1337 0.1427 0.1358 0.1433
1
共2兲_{共JT兲} _{0.0765} _{−0.1559} _{−0.1507} _{−0.0384} _{−0.1332} _{0.0347}
1
共3兲_{共JT兲} _{0.6423} _{0.2933} _{−0.3355} _{0.1271} _{−0.7872}
1
共4兲_{共JT兲} _{1.0700} _{1.5370} _{−3.7935} _{1.4357} _{1.9497}
2
共4兲_{共JT兲} _{0.5387} _{0.4516} _{2.5467} _{0.7422} _{−2.3722}
1
共5兲_{共JT兲} _{−1.3896} _{2.2688}
2
共5兲_{共JT兲} _{7.5791} _{8.1847}
1共6兲共JT兲 97.0116 −13.1908
2
共6兲_{共JT兲} _{−55.8532} _{62.5649}

*FIG. 3. CI coefficients along x from reorthogonalized 2*⫻2 CI coefficient
matrix obtained in the diabatic orbital basis. Main configurations
. . .*共1ex*兲2*共1ey*兲2*共3a*1兲1of2*A*1state and . . .*共1ex*兲1*共1ey*兲2*共3a*1兲2of2*Ex*state

coin-cide共open circles兲; contribution of main2*A*1configuration in
2

*Ex*state共open

triangles兲 and of main 2*Ex*configuration in

2

*A*1state共open diamonds兲 are

representations of the JT and PJT problem for this system,
which yield diabatic states in agreement with the above
analysis, are obtained from the JT6/PJT4 and JT4/PJT4
com-binations. Both models result in similarly small fitting errors.
The effect of the orders of the diabatization scheme on
the nuclear dynamics is investigated by wave packet
propa-gations. The dynamics after a vertical excitation onto one of
the two excited diabatic surfaces has been studied using a
standard propagation scheme based on Fourier
transforma-tion. The initial Gaussian wave packet was centered at
*共x,y兲=共0,0兲 with a width of 40.8 a.u. A constant mass of*
1943.38 amu is assumed for the kinetic part of the
Hamil-tonian, leading to an energy of around 5.6 eV for the wave
packet. Figure 4 presents the evolution of the ground state
adiabatic electronic population for different orders of the
model. Large differences are observed during the first 50 fs
with a much slower population transfer to the ground state
for the lower PJT order 共models JT2/PJT1 and JT4/PJT1兲.
The populations obtained with the higher order models JT4/
PJT4 and JT4/PJT6 are similar and only differ in details. For
longer propagation time 共up to 500 fs兲, oscillations of the
population around 30% for the model 21 and 40% for the
other models are observed. The differences obtained for this
low dimensionality problem共two vibrational degrees of
free-dom on three coupled electronic states兲 indicates that larger
differences will be observed for higher dimensional systems
because of the increased possibility for the energy to flow
from one vibrational mode to another one.

**V. CONCLUSION**

In this paper, the diabatic Jahn–Teller and pseudo-Jahn–
Teller Hamiltonian matrix is presented up to sixth order in
both the complex and real representation for a general
*sys-tem with a C*3symmetry axis. The different possible

symme-try groups *共D _{3h}, C*

_{3}

_{v}, C_{3h}, and C_{3}兲 and coupling mode sym-metries

*共e*

### ⬘

*and e*

### ⬙

兲 have been investigated. Furthermore, the*dynamical pseudo-Jahn–Teller effect is discussed. In such a*

case, the strength of the PJT coupling depends upon distor-tion along an addidistor-tional coupling mode.

Application to the representation of a two-dimensional
cut of the ground and first exited state of the ammonia cation
is presented. Based on the analysis of both the fitting errors
*to the adiabatic ab initio energies and of the evolution of the*
*CI coefficients along the x vibrational coordinate, the *
com-bination of Jahn–Teller coupling up to sixth order and
pseudo-Jahn–Teller coupling up to fourth order is found to
be the most suitable one. It is shown that the higher order JT
and PJT expansion developed here can represent the strongly
anharmonic potentials of the model system with excellent
accuracy. The analysis of dynamical properties and, in
par-ticular, of the adiabatic populations underlined the significant
role of the coupling order. Application to the NH_{3}+system in
full dimensionality is in progress.

**ACKNOWLEDGMENTS**

The BFHZ-CCUFB is acknowledged for financial
sup-port. The authors thank Alexis Viel for checking the
calcula-tions. Stefanie Neumann is acknowledged for assistance with
*the ab initio calculations.*

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22

**M. Döscher, H. Köppel, and P. Szalay, J. Chem. Phys. 117, 2645**共2002兲.

23

The convention used in Ref. 20 is兩1典=兩*x*典 and 兩2典=−兩*y*典.

24

MOLPRO*is a package of ab initio programs written by H.-J. Werner and P.*
J. Knowles, with contributions from R. D. Amos, A. Berning, D. L.
*Coo-per et al..*

FIG. 4. Adiabatic ground state population as a function of time in femto-seconds for the models JT2/PJT1共dotted line兲, JT4/PJT1 共dashed line兲, JT4/ PJT4共dot-dashed line兲, and JT4/PJT6 共full line兲.