Reaction holonomy and the geometric phase of quantum evolution in chemical reaction systems

THEO CHEM ELSEVIER

Journal of Molecular Structure (Theochem) 310 (1994) 1-11

Reaction holonomy and the geometric phase of quantum evolution in chemical reaction systems Akitomo Tachibana Division ofMolecular Engineering, FacultyofEngineering, Kyoto University, Kyoto 606-01, Japan

(Received 19July 1993; accepted 31 July 1993)

Abstract A new geometric phase associated with the reaction holonomy is found where the vibrational motion along the reaction coordinate brings about rotation of the reaction system. The interplay of this new phase with the Berry phase is disclosed.

1. Introduction In the quantum mechanical treatment of chemical reactions, the essential background lies in the Hilbert space of wavefunctions where the coefficients of wavefunctions are in general complex numbers. Recent progress concerning geometric phases in physics deals with the "phase factor" of wavefunctions, the Berry phase [I,2]. Consequently, it is of great interest to study' the geometric phase in chemical reaction theories [3]. The first observation of the geometric phase in chemical reaction theory goes back to the work of Hertzberg and Longuet-Higgins [4] and Mead and Truhlar [5], where the conical intersection occurs in the adiabatic change of the electronic state. The geometric phase appears when one keeps the electronic phase unchanged if an infinitesimal change of nuclear framework takes place. Mathematically, this "phase unchangedness" is the rule of parallel connection [6]. A new geometric phase associated with the reaction holonomy [7] has been found where the

vibrational motion along the reaction coordinate should inevitably bring about rotation of the reaction system [7,8]. The interplay of this new phase with the Berry phase is disclosed.

2. Wavefunction of a chemical reaction system

2.1. Adiabatic expansion The conventional approach to the dynamics of an isolated nonrelativistic chemical reaction system utilizes the orthonormal complete set {~/}/=I.",.oo of adiabatic electronic wavefunctions, and expands the total wavefunction Wtotal as follows: 00

WtotaI

= exp(i,dtota\(t)) LWnuc/~/

(2.1)

/=\

with (2.2)

where the expansion coefficient

0166-1280/94/S07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0166-1280(93)03523-A

wnuc/

serves as the

A. Tachibanaj.I, Mol. Struct,(Theochem) 310 (1994) 1-11

2

wavefunction for the nuclear motion. The total wavefunction 'l'total satisfies the time-dependent Schrodinger equation: . d'l'(t)

Itld!= Htotal'l'(t)

The matrix element of the total Hamiltonian Htotal of the reaction system is then given by [9] (('l'dTnue + He + Vnucl'l'2» =

(2.3)

(2.10)

(2.4)

t.:

where is the kinetic energy of the nuclei involved in the reaction system, He is the electronic Hamiltonian, and Vnuc is the nuclear repulsion potential. The total dynamical phase 'Ydtotal(t) is given as

'Ydtotal(t) = -

*r

(('l'total(t') IHtotalI'l'total(t')))dt'

(2.5)

using

£_1 8wj. 8W2 = 0=

111/0

8xa

8xa

2:)'l'nuc/(t)I'l'nuc/(t))= 1

(2.6)

1=1

The rotational and internal degrees of freedom of the nuclear framework can be further treated using fibre bundle theory [9]. The nuclear wavefunction 'l'nucl can be classified according to the irreducible unitary representation of SO(3) for the rotational motion. Using the Ith order rotation matrix D 1, satisfying

(2.7) the nuclear wavefunction reduced from Eq. (2.1) to

(I-I)ab(Ja'l'j)(Jb'l'2)

f

+L

where the ((» denotes the double integrations with respect to the electronic degrees of freedom and the nuclear degrees of freedom using the normalization conditions (Eq. (2.2» and 00

t

a,b = I

aij(~iWi)(~jW2)

ij=1

(2.11) where WI and 'l'2 denote arbitrary wave functions, Xc = (Xc, Yc, Zc) denotes the Cartesian coordinate of the centre of mass of the nuclei, X a denotes the Cartesian coordinate of the nth nucleus in the centre-of-mass system X together with its mass 11/0' and I is the tensor of inertia of the nuclear framework with respect to X . The Ja denotes the rotational vector and ~i denotes the vibrational vector of the ith internal local coordinate qi (i = 1, ... ,J) of the nuclear framework, both of which will be given in Section 2.3. The volume element aside from the electronic part takes the form

dV = sin ej}JintdXC /\ dYc /\ dZ c /\ d1

I I I 'l'nucl(t) = VI 2 "L.- DJlm'l'nuclmJl(t) 87f Jl"=-l

(2.8)

1 .21 + 1

00

I

(2.9)

1=1 Jl=-I

where 'l'~uClmJI(t) represents the wavefunction for the nuclear internal motion of the reaction system.

1/2

a

I

1

L L ('l'~uclmJl(t)l'l'~uClmJl(t)) =

N )3 L/1/

0; ( II

with the normalization condition

(2.12)

/\ d2 /\ d3 /\ dq' /\ ... /\ dqf

det II lab

II det II aij II

(2.13)

11/0

0=1

where a (a = 1,2,3) are Eulerian angles [10] and lIa;jllis the inverse of the Wilson's G matrix [11] for nonrigid molecules [9].

A. TachibanajJ , Mol. Stru ct, (Tlzeoclzem) 310 (/994) 1-11

2.2. Geometric phase

where h(1) is the one-electron Hamiltonian, and

2/r12 e is the electron-electron repulsion potential.

A geometric phase associated with the change of the nuclear framework is presented. Here nuclear coordinates which depend on time are treated classically. Then the time evolution of the electronic wavefunction (t) is given as

itt d~~t)

= He(t)(t)

(t) = exp(hd(t)) exp(i'Yg(t))o(t)

(2.15)

{r

(o(t)Io(t))

(2.20) where P denotes the linear momentum vector of the "collective" translation of the nuclear framework. The P is obtained as follows:

P=~

(o(t')IHe(t')Io(t'))dt'

= I

(2.17) The geometric phase 'Yg(T) is obtained by integrating the following differential equation from t = 0 to t = T:

d'Yg(t) = i( (t)1 do(t) ) dt 0 dt

(2.18)

It should be noted that (t) is in general a wave packet and Eq. (2.17) need not be satisfied, then the Aharonov-Anandan phase [12] results. If it is restricted to be an eigenfunction of /le(t) for any t, then 'Yg(t) reduces to the Berry phase [I]. 2.3. Electronic Hamiltonian

The electronic Hamiltonian He is given in the second quantized form as

Ja+(I)h(l)a(l)dvl +~J Ja+(I)a+(2) ~: a(2)a(l)dvl dv

(2.21) .

aXe

(2.16)

Consider a nuclear framework at t = O. If the nuclear framework in X changes its shape and returns to its original shape at a later time t = T, namely traces a closed trajectory, then we have

He =

The field operators of electrons are denoted as a( I) and 0+(1) , which are integrated with the volume element dVI over space and spin coordinates to give the He. First, note that the He is invariant under the "collective" translation of the nuclear framework:

(2.14)

The geometric phase 'Yg(t) in (t) is obtained along with the dynamical phase 'Yd(t):

'YAt) = -

3

Thus equation (2.20) allows us to treat the electronic structure in the centre-of-mass system X of the constituent nuclei. Second, note that for any shape of the reaction system, the He is invariant also under " collective" rotation of the nuclear framework:

[J,He ] = 0

(2.22)

where J denotes the angular momentum vector of the "collective" rotation of the nuclear framework. The rotational vectors Ja which are the components of J with respect to X have been obtained as [9] J = a

~(lrl)b~ Z:: a 8A-.b

b=1

(2.23)

'I-'

Equation (2.22) further allows classification of the electronic wavefunction according to the irreducible unitary representation of SO(3). The electronic Hamiltonian He is dependent only on the residual degrees-of-freedom of the reaction system, namely the! = 3N - 6 vibrational degrees of freedom , N being the total number of nuclei involved in the reaction system. Indeed, the He is not invariant under vibration of the nuclear framework: (2.24)

2

(2.19)

where ~i denotes the vibrational vector of the ith internal local coordinate qi (i = 1, . . . ,f) of the

A. TachibanajJ, Mol. Struct, {Theochem) 310(1994)

4

nuclear framework. The vibrational vectors been obtained as [9]:

~;

a

= fjiq

~;

have

a

(2.25)

There further emerges the vibration-electron interaction called the vibronic interaction [13,14,8].

2.4. Reaction 110/0110111)' It should be noted that even a small vibrational motion may bring about the rotation of the reaction system [9]. Indeed, the vibration-rotation coupling appears in the commutator of the vibrational vectors

[~;, ~J = -

3

L F/}Ja

(2.26)

IJ -

First, the matrix form of tI>(t) is given by Eq. (2.15) as follows:

tI>(t) = exp(iid(t» ~ 21 + I

(3.1 ) Here "~II(t) denotes the geometric phase with respect to the component ll~(t), which are so defined in Eq. (3.1). Because the rotation matrix D' is unitary, the normalization condition is 1

2/

af3f _ apt aqi

Bq!

I

+I

L S~Jl = I; Jl=-I

S:,V = (1l:,11l~);

S~I' = 1 (3.2)

(2.27)

The F;} can be interpreted as a gauge field on M and the Pia as the gauge potential. This is called the reaction holonomy [7]. The origin of the reaction holonomy is that the Eckart frame for the treatment of the rigid to semirigid molecular vibrational problems is not uniquely determined in nonrigid objects [15]. Differential geometry is used in order to obtain the internal coordinate space M. Since the translational motion of the system can be separated out, the centre-of-mass system X for nuclear configuration in the system can be obtained. In this space X, we let act the group G = SO(3) of rotations gx = {gx!>. oo,gxN};gE G,x E X

(t.1 D~m(t) exP(ii~Jl(t»ll:,(t»)

x

a=1

with F,.'! _

3. New geometric phase

3.1. Differential approach

3

L fJtJ a=1

i-n

The dynamical phase ld(t) is given as

ld(t) =

I

I

-h 2/ + I x

i:r

Jl=-I

(3.3)

(ll:,(t')IHe(t')lll:,(t')}dt'

It should be noted here that the rotation matrix

D' is common for both the nuclear framework and the electronic structure attached to it. The differential equation for l~I,(t) is found to be

d"L(t) . ~ I I d t = I LJ wl'vSJlVexp(.l([gVI v=-I

(2.28)

I ) 19l' ) (3.4)

Then an individual nuclear configuration corresponds to a point in the quotient space

M=X/G

(2.29)

In differential geometry, X is treated as a principal fibre bundle over the base manifold M with structure group G = SO(3). This abstract manifold 111 is referred to as the internal space of the system [9]. The connection form associated with the vibrational motion is then introduced [9].

I I = "'" wpv LJwmpv

(3.5)

m=1 I

_ Dol

wmJl v -

um

dD~m dt

(3.6)

where w~v is identified with the connection form introduced by Iwai [16,8]. It should be noted that

A. Tachibanall, Mol. Struct. (Theochem)3/0 (1994)

the nonlinear contribution of ')'~J.' (t) is present in the right-hand side of this differential Eq. (3.4), distinct from the former differential Eq. (2.18). The mth component of the connection form W~'IJV is obtained by using exp( -ijlq})dfu,,(¢2)

1

1

~

d~l)

+ exp( -ill¢l)

di

(¢2) :i¢2

3 8D1 L 8 n~v (O-I)'bf3ib a=1 b=1 ¢ 3

dr/i ) x exp( -imr/}) ( dt

= - L

3

= - L

a=1

+ exp( -ijl¢1 )dfu,,(¢2) X

exp( -im¢3) ( -im

3

d~3)

f

[9]. If the derivative with respect to time is restricted as (3.9)

then the differential equation for ')'~J.'(t) reduces from Eq. (3.4) to

Lf [ 8"1gJ.'~t) 1

Bq' [

]

1

+ i(<1>/\ l'

8<1>~ )] dqi 8q' dt

In the differential approach, the geometric phase depends on the Eulerian angles. If the invariant integral on SO(3) is used, then the geometric phase that is independent of the Eulerian angles can be obtained. Let the volume element be given as (3.14) where Ilg denotes the SO(3)-invariant Haar measure. Since the volume of the compact group SO(3) is 871"2, the normalization constant in Eq. (3.1) changes accordingly and Eq. (3.1) reduces to:

<1>(t) = exp(i"ld(t))yS;2~ 8n2 21 + I x

dq'. dt

i V~I w:J.'vS~v exp(i("I~v - "1:1'))

f3tJaD~lV

3.2. Integral approach A

0= LOCd¢b"+ Lf3iadqi (3.8) b=1 i=1 where Oe and f3t are determined by the connection form associated with the vibrational motion in M

~_ f dqi ~ 8t - ~ dt Bq!

(3.13)

where we have used Eqs. (3.8) and (2.23).

where dfu,,(¢2) are Wigner functions [10]. The Eulerian angles satisfy

f

(3.11 )

8D~v _ ~8D~lV 8¢a 8 qi - ~ 8¢a 8 qi (3.7)

r;

1

wiJ.'v = L..J WmiJ.'v m=-I

with

x exp(-im¢3) ( -ijl

=

5

(3.12)

dD~(t) =

i=1

t-u

CtlD~,(t) eXP(hiJ.'(t))<1>~(t)) (3.15)

(3.10)

The dynamical phase is again given by Eq, (3.3). It should be noted that the SO(3)-invariant integral is also utilized in Eq. (2.7), but the physical meaning is different: in Eq. (2.7), the integral was performed for the nuclear degrees of freedom, however here in this and Section 3.3, the integral is for

A. TachibanalJ.Mol . Struct. (Theochem} 3/0 (1994) 1-11

6

the electronic state. Then the differential equation for 1~J'(t) reduces from Eq. (3.10) to

.( I

I))

xexp (1 I gmv -Igm/l


+ 1.

118ll>1' ll>J' 8 i q

dq'

(3.20) where (W~Jil"jg is given by Eqs. (3.18), (3.12), and (3.13).

3.4. Adiabatic electronic wavefunctions

(3.16) I

(wLv}g =

L

(W~i/lv}g

(3.17)

m=-l

J

I I (WmiIW}g = 87rI 2 wmi/lVdJLg

If strictly adiabatic separation of the electronic motion with respect to the change of the nuclear framework is imposed, then ll>~(t) will be independent ofI and JL: namely, an orthonormal complete set {ll>[}[ = 1,...,00 of adiabatic electronic wavefunctions, Eq. (2.2), is adopted. So that (3.21)

(3.18)

where W~iJ'v is given by Eqs. (3.12) and (3.13). Now that the integration over SO(3) is performed, the geometric phase is independent of the Eulerian angles.

3.3. Integral approach B If the quantum number 111 is also conserved, a more precise geometric phase may be obtained as follows :

and then (3.22) Then Eqs. (3.4), (3.10), (3.16), and (3.20) are simplified. It should be noted that even in this simplified level, the geometric phase is composed of two contributions, the interplay of the new phase with the usual Berry phase.

4. Vibronic Hamiltonian I

X

L

(3.19)

D~(t) exp(il~mJ'(t))ll>~(t)

The field operators a(1) and a+(1) of electrons are commutable with P, 1 and ~i:

/l=-l

where I~ml,(t) denotes the geometric phase with respect to the component ll>~(t). The dynamical phase is again given by Eq. (3.3) . The differential equation for I~m/l(t) reduces from Eq. (3.16) to

[P,a(I)] = [l,a(I)] = [~i>a(I)] = [P,a+(l)] = [J,a+(1)] = [~i,a+(1)]

=0

(4.1)

Let the orthonormal complete set of spin orbitals

1/1i( I) for the representation of the field operators a(l) and a+(I) be given as follows: 00

a(l) = Lai1/1i(l) and i=1

00

a+(l) = Lat1/1i(l) i=1

(4.2)

A. Tachibauaj.I, Mol. Struct,(Theochem) 310 (1994)

Since the orbitals are defined in X, we have

[P,1Pj(1)] = [J,1Pj(1)] = [P,1/1j(l)] = [J,1Pj(I)] =0

(4.3)

but (4.4)

t-u

7

Hamiltonian and the dressed nuclear Hamiltonian will be given. The principal result is that the time-reversal pair of electrons exhibits special attractive interaction [14]. For stationary states of time-reversal pairs of electrons represented by real orbitals, the vibronically dressed electron-electron interaction is given as follows:

Equation (4.4) is the representation of the orbital vibronic coupling. Accordingly,

[P,aj] = [J,aj] = [p,aj] = [J,aj] = 0

(4.5)

(4.14)

but

(4.6) Performing a unitary transformation results in the dressed vibronic vector Sj

(4.15)

(4.7)

(4.8)

K;(l,2) =

1

00

2L{[61h (l )]1/1j (2) ;=1

(4.9)

-1/1j(1)[';;1/1j(2)n The dressed vibronic vector S; satisfies the same commutation relationship as the original one

(4.10)

[S;, Sj]

= [~i> ~j] = -

3

L F/'jJo

(4.11)

(4.16) where f denotes the operator for which the eigenfunction is the orbital 1/1; and the eigenvalue is the orbital energy c;. Here the spatial part of the orbital is written as 1/1;. Qndenotes the normal coordinate. It should be noted that the vibronic integral A (2) WIij} is positive definite if it satisfies the selection rule

0=1

and moreover should be "transparent" with respect to the field operators

lSi, aj] = [3;, ail = 0

(4.12)

Using the dressed vibronic operator (4.13) The procedure of Ref. 14 is followed and the complete vibronic Hamiltonian that exhibits the friction effects both in the dressed electronic

(4.17) Then the nth normal mode induces the vibronic attraction as shown in Eq. (4.14). The left-hand side of Eq. (4.17) is referred to as the extended orbital vibronic constant (EOVC). This attraction is immaterial if the orbital energy difference is large, but is significant if (4.18)

A. TachibanalJ. Mol. Struct. (Theochem} 310 (1994)

8

If the residual disturbance effects are to be included, the energy denominator is renormalized [14]

I cr,/3 IT _ Y pair -

2 '""'" L,;

00

'"""'(("j"} A(2)("I"}} + + L,; lJ lJ - Ll. lJ lJ ai(Jaia'aja'aj(J

(J#(J' ij.= I

(4.19)

i-n

degenerate t lu orbital. The electronic configuration of C~ is obtained as follows: tl u x tl u = lAg +3T lg + IH g

(5.1)

For the triplet state, the H mode can resolve the degeneracy: (5.2) For the singlet state, the G + 2H modes can resolve the degeneracy:

(4.20) where W n denotes the frequency of the nth normal mode. The disturbance effect may in some cases work as the resistance for those electrons in the vicinity of the Fermi level for which lEi - Ejl may be smaller than W n • It should be noted that this is the disturbance effect for Cooper pair formation: indeed, the attractive force is given by Eq. (4.16) in a closed form, and the residual interaction in Eq. (4.20) is to inhibit the attractive force if

lEi - Ejl < W n • These are an extension of the similar attractive force for Cooper pair where the translational symmetry of the crystal is essential to define the electronic state in terms of crystal momentums [17,18]. It is to be noted that the Cooper pair defined in infinite systems is considered the special limit of the time-reversal pair defined in finite systems. Recent work of the electron correlation in the scattering continuum demonstrates that the stabilization mechanism of the time-reversal pair is also present [19].

IC~O : [Hf = A + G + 2H

(5.3)

First, for triplet state, the T I ® h instability problem is treated: for 3C~O, [TIl 2 = A + H. The vibronic coupling operator is given as V(Q) = E h VHh(QHh)

(5.4)

with

vljh(QHh) = QHh x FH x (TliIHhTIj)

(5.5)

where the H mode is five-fold degenerate: {QH9' QHfl QH~, QHl)' QHd· The Jahn-Teller energy is given as

The Yc,(Q) are the eigenvalues obtained as det IlVij(Q) - Yc,(Q)bij II = 0

(5.7)

According to the procedure [21], the state vector is represented as IT lcr > = xlT lx > + ylTly > + zlT lz >

(5.8)

The potential energy surface is given as

where U is the vector of the electronic state: 5. Geometric phase in the doped C60 The Jahn-Teller instability of C 60 is an interesting example of the geometric phase where the reaction coordinate which connects many local minima exists within a small sphere. Ceulemans and Fowler [20] have studied the Jahn-Teller instability of C 60 in terms of group theory, but the phase argumentation has not been performed. C 60 has icosahedral symmetry (Ih) and the lowest unoccupied molecular orbital (LUMO) is a triply

U = t(x,y,z)

(5.10)

The extremum on the unit sphere U is then found to be a simple paraboloid

Eo -EJT H

(5.11 )

where

Ell =

-(1/3) x (F~I KH )

(5.12)

This shows the Mexican hat Jahn-Teller instability.

A. Tachibana[J, Mol. Struct, [Theochem] 310(1994) 1-1I

Second, for the singlet state, the H ® (g ~ 2h) instability problem is treated: for IC~, {Hf = A+G+2H . The vibronic coupling operator is given as

V(Q) =

'BAAVAA(QAA)

(5.13)

9

(5.25)

Ef~ = -(2j5)(FflbjKH)

(5.26)

The important relationship is found to be

FHb = (J5j3)Fll a

with

vGgij(QG,) = QG, x FG x (H;lGgHj )

(5.14)

where the G mode is four-fold degenerate, {QGa, QG.o QG)" QGz}, and where

VHhij(QHh) = QHh

X

[FHa X (H;iHahH)

(5.27)

Now, the molecular orbital instability of C60 is found to be (5.28)

(5.15)

(5.29)

+ F Hb x (H;IHbhHj)J The potential energy is then treated in totally ninedimensional coordinate space. The Jahn-Teller energy is given as

with

Eo(Q) = (lj2)E AAKA X QXA + Va(Q)

(5.16)

Note that the EOVC (extended orbital vibronic constant) introduced by Eq. (4.17) is here given as

(5.17)

Ji~{ = (l/J/1i18fj8QHhltPrlj)

The Va(Q) are the eigenvalues obtained as det \I J'ij(Q) - ~(Q)8;j \I = 0

(5.30)

=

The state vector is represented as IHo> = BIHo > + EIHf > + ~IH{> +1JIHI) > +

(5.18)

The potential energy surface is given as

Ea(V) = (lj2)EAAKA x

QL + 'V· V· U

(5.19)

where V is the vector of the electronic state:

The extremum on the unit sphere V is then found to have three characteristic points:

= E~\

(signature of Hessian

= 2)

(5.31)

(lliIHhtlj)

The selection rule for the vibronic attraction is the non-zeroness of the EOVe, which has been calculated for the present problem [22,23]. The state vector is represented as

IlIa> = altl"-> +blt l)'> +e/tl z>

(5.32)

The Jahn-Teller energy is given as (5.20)

ED5d

fu x

(5.21 )

Ea(Q) = II/(Q). ir'o» + v(Q» 'I/(Q) where II

II

(5.33)

is the vector of the electronic state

= l(a,b,e)

(5.34)

The Ea(Q) are the eigenvalues obtained as

ED2h = E:~ + (lj6)E{J (signature of Hessian = I) (5.22)

EDJd = E:~ + (4j9)EbT (signatur e of Hessian

=

0) (5.23)

i~d(Q)8ij

11=0

(5.35)

Thef is diagonal in this representation, so that the problem reduces to (5.36) The important relationship is found

where the Jahn-Teller energies are given as

E{J = -(lj2)(F~jKG)

detllJir(Q) -

(5.24)

fll = -.j(2j3)FlIa

(5.37)

A. TachibanalI , Mol. Struct. (Theochem) 310 (1994) I-ll

10

The matrix elements Vij(Q) are represented in the unit v = (2/9)F~a/KH(= IE~1\) as follows: v(D Sd ) = v

(~

o

3/~)

-3/2Js - 1/2 3/.;5 3/2.;5 - 1/2 (5.38)

JJ

-I) -I

(5.39)

V2Z =

vsin(¢ + az),

(5.46)

V2

=

V3

= V33 = -(23/32)\/zvsin(¢ + a3),

with sin

a\

= (3J5 - 1)/8,

sin az = (3J5 + 1)/8,

sin

a3

= -1/4J46,

cos

a\

=

-(vI5 + v3)/8, (5.48)

cos az = (v15 - J3)/8, (5.49)

cos

a3

= 7vI5/4J46.

(5.40)

o

(5.47)

(5.50) The corresponding coordinates are given as

The eigenvectors are found to be

(5.51 ) (5.41 )

(5.52)

(5.42) QHO

{V/l«D 3d)}

= {-2v,v,v}

- v(I/8)(FH b / KH ) sin ¢

(5.43)

The eigenvalues together with the symmetry are schematically shown as follows:

= -v(3/8)(FHa/ KH ) cos ¢

QHf

= J(3/8)(FHa/ KH) sin ¢

- v(I/8)(FH b / KH ) cos ¢

b3u

(5.53)

(5.54)

(5.55)

It is easily proved that the diagonalization is confined within the functional space spanned by IHO) and IH€), then we obtain the great circle [20] which traverses the centre line of the collective pseudorotation path [22,23]. If () = cos(¢!2), € = sin(¢!2), so that

IHa > = cos(¢/2)IH o >

+ sin(¢!2)IH, >,

(5.44)

then we find (5.45)

Along this great circle, the orbital energies return to their original values. However, clearly, the electronic state changes its sign + --> - as we traverse one cycle along the great circle. Then the single-valued electronic wavefunction can be obtained [4], where the nonzero contribution of the form

is significant for the Berry phase [1-6]. It should be clear that the electronic adiabatic wavefunction gets the geometric phase not only from this one but also from the reaction holonomy, yielding ,.riw In this connection, it is well known that the eGO sphere is rotating rapidly in the condensed

A. TachibanaiJ, Mol.sira«. (T/zeoclzem) 310 (/994) 1-11

phase, so that the interplay of the new geometric phase with the usual one should be very interesting. The contribution of the geometric phase to the vibronic Hamiltonian is under progress and will be published elsewhere [24].

Acknowledgements This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan, for which we express our gratitude. The author thanks Professor T. Iwai for his kind discussion on Ref. 16.

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