Analytic treatment of the vibronic coupling problem

Chemical Physics 59 (1981) 279-287 North-Holland Publishing Company

AN!&YTIC Sighart

F.

TREATMENT FISCHER

Physik-Department

Received

PROBLEM

and Horst KePPEL*

der Technirchen

15 January

OF THE VIER6M’dIC COUPLING

UniaersitZt

Miinchen,

8046

Garching,

West

Germany

1981

A model system consisting of three electronic states kd one vibrational degree of freedom is investigated. The upper two electronic states are vibronically coupled. Using the method of canonical transformations and avoiding the adiabatic separation between nuclear and electronic motion, approximate eigenfunctions are constructed and analytic expressions for the eigenvalues are derived. The energies and the transition matrix elements between the ground state and excited states are compared with exact numerical solutions of the problem.

1. Pntroduction The interpretation of molecular structure is usually based on the adiabatic approximation. In particular, the concept of potential surfaces for the nuclear motion is a direct consequence of this approximation_ The nonadiabatic coupling is commonly neglected or treated within low-order perturbation theory. For most situations this is suflicient, but there are several phenomena which cannot be accounted for in this way. A typical example is the so-called Jahn-Teller distortion [l], which describes the change of the equilibrium configuration of a molecule. It occurs if the components of a degenerate electronic state are vibronically coupled. Another example is the appearance of mystery bands in the photoelectronic spectrum of small molecules [2]. They are due to the combined coupling effect of totally symmetric and non-totally symmetric modes. Finally, we want to point at a situation observed for naphthalene [S] and ovalene [4], which is typical for many high-resolution spectra. In naphthalene vlbronic bands belonging to the second excited state (52) have been analyzed with high resolution [S]. It has been found that vibronic bands of St contain very detailed structure, which is due to mixing with vibronic states from the first excited state (S1). In order to understand the location and intensity distribution of such bands, perturbation theory is unsatisfactory and thus the whole concept of potential curves belonging to the different electronic states is no longer very useful. Alternatively we shail classify vibronically coupled electronic states with their vibronic manifolds according to their symmetry only. Such an approach has been first proposed by Fulton and Gouterman [6] in connection with the dimer problem [7]. Our aim shall be to derive analytic expressions for the spectrum and the transition probabilities from the ground state into such coupled vibronic states. We concentrate on the simplest model system consisting of two electronic states coupled by one vibrational mode only. We want to demonstrate that the method of canonical transformations can be a very useful tool in this fieid. The analytic results involve of course approximations. In order to test the range of validity, we compare the results with exact numerical calculations, which are easy to handle for this simphfied model [2, S]. The analytic method can be extended to several modes, which become quickly untractable for the standard diagonahzation procedures. Applications to larger molecules such as naphthalene or pyrazine for which it is known

* Presentaddress:Institut

fiir Theoretische

030I-0104/81/0000-0000/$02.50

Chemie,

0

UnivtrsitSit Heidelberg,

North-Holland

6900

Publishing

Heidelberg,

Company

West Germany.

S.-F. Fischer, H. KiipFel/ Analytic treatment of the uibronic coupling problem

280

that the vibronic couplings influence the radiative and nonradiative will be presented

in forthcoming

transitions

in an unusual manner

papers [Y].

2. The model hamiltoniax We consider three electronic states, &(7, q). i = 0, 1,2, where r stands for the electronic coordinates and q describes the nuclear degree of freedom with equilibrium position in the electronic ground state qO_We assume further that the nuclear potential belonging to the electronic ground state is harmonic and that the two excited states are coupled linearly in the nuclear coordinate. The hamiltonian for this system has been derived previously [2, lo] and can be written as

o is the frequency, b and 6; are the Bose-type creation and destruction operators for the vibrational excitation in the electronic ground state. Eo, El and Ez are the electronic energies including the zero-point energy of the vibration. A is a dimensionless coupling constant. Later we shall also use the dimensionless energy E = (E2 - El)/Zzo. Now we split up the hamiltonian into two noninteracting contributions with the help of the following canonical transformation: H,=

U:HU,,

(2)

and G = exp (i&-b)

(4)

and obtain:

(6)

h, =b’b+~(b+b)+fa~G. Here the foxowing properties

GbG =-b, The The term In

of the transformation

of b have been used:

G’=l.

(7)

separation of H, ida electronkalIy noninteracting contributions reflects an internal symmetry. price one has to pay fcr ihis decoupling is a nonlocal anharmonic potential represented by the &z&G. the following we shall try to find approximate solutions for the eigenvalue problem of h,.

3. The canonical

IransformatiIon

The hamiltonian h, contains the nmmber operator SCb, the term linear in b’ib which is nroportional to the cou$ing parameter A and the nonlocal potential proportional to --E.If E is..small 2ompared to one ar,d compared to A, we may neglect it in lowest order. This is the Jahn-Teller limit and the hamiltonian can easily be diagonalized with a canonical transformation wbich~describes a

/ Analytic treatment of fhe Gbronic couplbzg problem

S.F. Fischer, H. K&d

displacement

of the coordinate

281

4

g=b--h

(8)

such that

~+6+A(~++~)=b+S-A2. This transformation

(9) can be generated

by the displacement

operator

U,, = exp (A (b - b +)I.

(10)

In this limit (E = 0) the spectrum eigenstates generated by b’ and order of magnitude as h or even additional nondiagonal elements general transformation

is shifted by -hoh’ and the overlap integrals between the those generated by b”’ form a Poisson distribution. If E is of the same larger this transformation alone would not help. It would generate in the transformed G-operator. We therefore introduce a more

(11)

U, = exp CQlbN exp IQh)l, with

QI(LY)= r,G(b++bL

Q2(a)=s,(b-6+).

(12)

The parameters r, and s, will be determined in such a way that the transformed first-order harmonic oscillator states. To transform the hamiltonian, the following relations can be used: exp [Q~(a)l=(b

exp [-QI(a)](6++6)

exp [-QI(~)IG exp C-Q&

exp [Q~(a)l=

eigenfunctions

are in

+-6’) exp C2Q~(cr~l,

G exe C~QI(~)I,

(13)

116 exp CQrb )I = b - si,

exp E-Qz~Y)IG exp CQAr)I = G exp C2Q2b)l. We thus obtain after some rearrangements h, = U,‘h,U,

=&$,,

+A(&

+&)+&YE&

=b+b+s~+r~-s,(b’+~)-r,Gexp[2Q~(cr)](6-b+)+h(b’+b-2s,)cos[2r,(b++b-2s,)] +~(~~sin[2r,(b~+b-2s,)]+h(b’+b-2s,)Gexp[2Qz(cr)]sinC2r,(b’+6-2s,)] &EG

exp [~Q~(cx)] cos [2r,(bi+b

-&)I.

(14)

It is further convenient to present the hamiltonian in a quasi-normal ordered form. That means, all destruction operators stand right with respect to the creation operators, only the G-operators are kept on the left side unchanged: h, =b’b+(sZ,+r~)-s,(b’+b! texp

(-2rt)

texp

Qir,b’)

-r-G

exp (-2sz)[exp

+exp (-2s:

exp (-4

is,r,){+A(b++2ir,)

exp (2ir,b)[_h(b

+$iA(b++2Er,)

-2s,)-&ze)~+c.c.

(-2s,bi)

- 2r$)G{exp

exp (2ir,b+) exp (2ir,b)

exp (2s,b)b -(6’+2.s,)

[(2ir, -2s,)b+]

exp [(2ir, -2s,)6’]

exp (-2s~b‘)

exp [(2s, +2ir,)b](&

exp C(2S_ +2ir,)b]+cc.).

exp (2s,b)]

+$iAb)

S.F. Fscher, H. KiipFclj Analytic treatment of the uibronic coupIing pmblcm

282

So far no approximations have been introduced. The parameters s, and r, are still arbitrary. For the Jahn-Teller limit we found r, = 0 and f, = A by eliminating the linear coupling term [eq. (9)]. Having two parameters, we can eliminate the terms linear in b*+ b as well as those linear in G&+-b). We find ihese from eq. (15) and obtain two coupIed equations for s, and r,:

r-(45’, -l)+s,(ae--4Ar,)

exp (-2rZ,)=O,

s, i (4Ar2, -A - (YET,) exp (-2r2,)

(16)

cos (4rpsp) +4Ar_s_ ezp (-2rt)

sin (4r,s,)

= 0.

(17)

Due to the presence of oscillatin g functions these equations have many solutions. In certain limiting cases, however, we can pick the solution of physical interest. Let us consider for instance the case of large energy separation E > 1 >A’. In this case the eigenstates are determined to a good approximation by the undisturbed adiabatic potential curves. The parameters r, and s, approach zero as E goes to infinity. So we can expand r, and s, in a power series of E-’ and get s, = -As -Zf4A3E-4;

r, =-arhs-‘-ccrh(4h’-l)~-~.

(18)

If we include certain higher orders in E we get a solution which applies also to the intermediate coupling regime

r, =

LYE&/(1

+4As,).

(20)

We shall denote a situation characterized by the condition E > A2 as weak coupling limit. In fig. 1 we have plotted the spectrum (see also section 4) for the parameter values o = 1, A = 1, E = 5. From eq. (18) we predict the values s+l= s-~ = -0.034 and r+l -- -0.18 which compare well with the numerical solutions from eqs. (16) and (17): sA1 = s-~ = -0.033 and r+l = -r--I = -0.18.

(b) Absorption exact aoiutian

I

I

“=I

ht.

X=1

0.8

Absorption linear teliminated

EZ5 f

+ --0.r

2

I

6

8

I

!

-2

.:

0

4, = -0.033 r., = -0.18

2

---0-L

--

-0.8

and intensities for the absorption for the weak coupling limit. In (a) the exact numerical results are plotted. In (b) the results of the analytic theory are presented for comparison. The two strong lines correspond to the two 0-O tcan-

Fig. 1. Eigemalues

sitions Tom the vibrationti ground state of the electronic groundstate into the vibrational ground states of the two electronically excited states respective!y.

SJ? Fischer, H. &ippel/ Analytic treatment of fhe uibmnic couplingpmbkm

283

4. The energy specfmn In this section we shall be iarge!y concerned with the strong coupling limit. Of particular interest is the spectrum of the low vibronic states. We want to determine the parameters r, and S, by minimizing the vibrational ground state energy for the electronically excited molecule. As approximate vibrational eigenvalues we take the expectation value of the transformed hamiltonian formed with harmonic oscillator wavefunctions. We write (15) in the form (21)

h,=h:+h:

with Jz”,ln) = b’bln) = nirz), hh is our perturbation term, its matrix elements become small for a proper choice of the parameters r, and s,. The diagonal matrix elements of h, read Ez = (nlh,ln)=

n +r; es’, +(-l)“ra

+e-2~{L~_1(4r~)4Ar,

e-2S5[4s,Lf,-1(4s;)

sin (4r,s,)+L0,(4r~)[(2Ar,

+2s,L~(4~‘,)]

-&~a) sin (4r,s,)-2As,

+ (_I)” e-2(%+e) [L”,(4rz +4s~)(fa~ -2Ar,>-4Ar,LA_1(4rZ

cos (4r&u

+4sZ)],

where L”, and Lk are Laguerre

polynomials. For the ground state we get

E,” = .s%+ rz + e-“‘[(2Arm -&)

sin (4r,s,) -2sJ

(22)

cos (4r,s, j] (23)

+ ePZC2r,s, + e-Zcsa+n’($~.a- 2Ar,). Our new equaticns for r, and S, follow from the minimalization conditions:

aEO/ar, =o.

dE,,/ds, = 0;

(24)

The first of these conditions is fulfilled if eqs. (16) and (17) hold. Essentially it is the sum of these equations: s, +e-‘%[(4Ar2, --a)&~, -A) cos (4r,s,)+4Ar,s, +e-2Sz[r,

-4r,s:

- (LLES,-4Ar,s,)

sin (4r,sJ]

e-2S] = 0.

(25j

Interestingly one obtains the same condition if the operator G is also normal-ordered and the coefficient in front of the term linear in 6” + b is equated to zero. The second condition in (24) reads: r, + emzeaA + a&r, +4h (s’, - r?)] sin (4r_s-) + (gAr,s, -ass,) +e-ZSz[s,

+ (LfAr2a--asr,

-A)

e-*4

cos (4r,s,)j W)

= 0.

An approximate solution in analytic form can be obtained by linearization of the equations in r, and S, -A. Taking only the dominant terms for A*> I> E we get s, =A +[QEA~~~~*~(~A*+~)-~.~~~~]/[A*(~-&

ee2\‘)(4A’+3)],

(27)

r, =a~/(6A~t2A). The adiabatic potential curve in the strong conp!ing limit is a double well potential for the lower electronic state. The energy d%erence between the two lowest vibronic states corresponds to the tunneling splitting. We get the equivalent value from the energy difference of the two vibronic ground states A& = E&X = 1) -E&Z = -1). Substituting the results from eq. (27) we find to lowest order in E and in e-2xZ: A& = E exp (-2A *)_

(28)

S.F. Escher, H. Kiippel/ Analyric rreatmenf of the vibmnic coupling problem

284

1 2nd 2 the calculated energies from eq. (22) are compared with exact cakulations for the parameters w = 1, A = I, E = 4~0.2. The parameters r, and s, were determined by searching for the energy minimum, rather than solving eqs. (25) and (26). The agreement with the exact results is better than i%. Within the Eorn-Oppenheimer approximation such good agreement cannot be reached. Ln particular the energy splitting, eq. (28), goes correctly to zero as E goes to zero. This is not so within the Born-Oppenheimer scheme. The results for the eigenvalues [eqs. (22) and (27)] apply also reasGnabIy well to the intermediate coupling Iimit E = A2 = I (see also fig. 3).

In tzhles

5. Transition probabilities In this section we want to evaluate the intensity distribution for the absorption and the emission spectrum. The matrix elements for these transitions stem from the transition dipole operator x. If & and & represent the exact eigenstates, the matrix element of interest can be written as:

(29) We can represent the dipole operator in the basis of the electronic states $9, +r, and $2. Assuming that there are only nonvanishing matrix elements between the ground and the excited states we get

We have seen, eq. (ai), that a second part which contains perturbation theory we may terms of the transformation

the transformed hamiltonian can be split into a non-interacting part and the transformed nonlocal potential. Within the frame of low-order neglect again the perturbation for the transformed wavefunctions. In operators U, and U, we can express this approximation in the form (31)

For +r we have the equivalent (29) we obtain:

exact representation

I&) = ~rn)~~& Substituting

these expressions

in eq. (324

Table1 Eigenvalues

TabIe2 Intensities

E ~0.2,w=h=l

E=-0.2,w=h=l

exact

exact

-0.9897

0.0385 1.0143 1.9683 3.0130 4.0121 4.9751 6.0218 6.9915 7.9927 9.0188

analytic 0.9881 0.0412 1.0217 1.9761 3.0180 4.0281 4.9886 6.0351 7.0168 8.0086 9.0455

analytic

-1.0168 --1.0153 -0.0423 --0.0399 0.4869 0.9945 2.0314 2.@381 2.9863 2.9907 3.9884 4.0045 5.0251 5.0364 5.9778 5.9913 7.0082 7.0322 8.0075 8.0215 8.9814 9.0085

E=-O.Z,o=A=l

&=m,o=A=l

exact

analytic

eXact

amlytic

0.3325

0.3425 0.3670 0.1966 0.0701 0.0187 0.0040 0.0007 0.0c0: 0.0000 0.0000 0.0000

0.4057 0.3665 0.1513 0.0561 0.0169 0.0028 0.0005 0.0001 0.0000 O.QOOO 0.0000

0.3939 0.3670 0.1708 0.0530 0_0123 Q.0023 d.0004 0.0000 0.0000 0.0000 0.0000

0.3608 0.2220 0.0668 0.0139 0.0034 0.0005 0.0001 0.0000 0.0000 0.0000

S.F. Fischer, H. K6pp.d / Analytic treatment of the uibronic coupling problem

28.5

and fif=~(x,,(m1(1-G)U-~ln}+Xoz~m!(l-tGfU+~in)}, If we choose m =0, R =X&(O~U_+zj’,

if

$t= $2.

Wb)

we get for+f=h;

for *f = *I.

=x’,,@\u+1i~>*,

(33)

In figs. 2-5 we plotted the transition probabilities proportional proportional to Xoz downwards in the same diagram. The matrix element (01 U,’ In}’ is easily evaluated: (O)UZ \n)” = c(r’D+s;)“/n!)

exp [-(6

i-s:)]{1

to X&

upwards and those

+sin 2[n arctan (7=/s,) -r,s,3).

(34)

In the first part we rediscover the Poisson distribution with a coupling parameter gz = rz + s$ This corresponds to the Franck-Condon factor of two displaced oscillators. The modification of this result

Tint. as

Absorption exact soluticn

Absorption ex0ct SO\ution

C!&

tI

Tfnt

Absorption Eg minimized IS=1 s.1 = 1.025 r4 =o.ot

x=1

drawnalwaysupwards,those cnrrespondiigto Q = -1 downward.

EO minimized

E-1

Fig. 2. Eigetwalues

and intensities far the absorption for the strong coupling limit. Again the exact and analytic results can be compared. Tmrdions mmeqwnding to Q = 1 are

Absorption

Int 0.8

I

Fig. 3. Eigenvalues and intensities for the absorptionfor the intermediate

strong aoupliting for the exact
analytic(b) theories_

SE Fischer. H. Kiippel/ Analytic treatment of the uibronic coupling problem

286

(al

Emission CxcCL YJlution

(a)

Int.

Emi ssian

exact solution

0.6

Ill:1

h=t

CO.1 k-1

1-n tronsiIiO%

E = 0.2

transition

0.L

a4

i

-L

-6

-2

J-?-I t 0

o-n

2

0

transition

++-+I

I

-0.L

OL

t 2.

/

(b) Emission

EOminimized

(bl

Emission to

Tint.

t

us1

1

minimized

Int.

0.8

a8 IA).1

A.1

l-n

E=a2

1-n

I.. 1 E= 1

1 0.L

i , 04

I c -6

!

’ -5

’

I‘-2

I 2

0

I -6

.!

,!

t L

:

I

2

AL

(0

I s+,4.t.t.,=a03fa t-n r, =O&_t=-O.O&On

-. -0.L S., =1.025.r~ =O.Olfor l-n tmnsitiom O-n ZL,=0375c~=-aOt.. 0-n ..

Fig. 4. The emission spectra from the vibrational ground states corresponding to (I = -1 are plotted downwards (Oe G transitions) and those corresponding to (I = +l are plotted upwards (1 -B n transitions). Again (a) and (b) refer to the exact and the analytic theory respectively.

{m1u, lo>*= [(r’, + ~t)~/nz!l

exp [-+l

+s%ilCl + (-1)“”

I

ltrar

..

:

:

1

itms ..

0-n -0L

i

Fig. 5. Emission spectra for intermediate coupling from the two vibrational ground states corresponding to a = 1 and a = -1. The exact results from (a) can be compared with the analytic results from (b).

is contained in the oscillating part. This part causes also a difference matrix elements for the emission

6. Discussion

2 o-n tmnsiticn

sin 2[m

arctan

compared (rol/sp)

to the corresponding

+ r&j).

of results

From the figures as well as tables it -an be seen that the method of canoni transformations works quite well. In particular the energy values agree within 1% with the exact ones in the strong coupling limit. The eneqq minimization procedure gives good results up to the intermediate coupling regime, where o = E = A’ = 1. In this case small deviations are found. The exact calculations predict the strongest intensity in absorption for the m = 0 to n = 3 transition among the odd states, while the analytic results have the highest intensity at the m = 0 to n = 2 transitions. +4lso the third odd-state should be higher in ener,gy than the.corresponding even state. For the emission spectrum the agreefnent is even better. The method of eliminating the linear terms works well in the weak coupling regime. The spectrum is characterized by very few lines separated by the energy gap E.

S.F. Fischer, H. Kiippetl AnaIytic heamenf of the uibronic coupling problem

287

The method can certainly be refined. For instance, the quadratic terms of the form bb + b’b’ can be eliminated by a unitary transformation involving the operator bb - b’b’. More interesting to us, however, seems to be the generalization to several normal modes [9]_ Acknowledgement The authors would like to thank Professors F. Metz and N. Heider for valuable discussions. References Cl] H.A. Jahn and E. Teller, Proc. R. Sot. A 161 (1937) 220. [Z] H. KGppel. Dissertation, Technische Universidt Miinchen, Germany (1979); L.S. Cederbaum, W. Domke and H. KBppel, Chem. Phys. 33 (1978) 319. [3] H. Gattermann and M. Stockburger, Chem. Phys. 24 (1977) 327. [4] A. Amiiav, U. Even and J. Jortner, Chem. Php. Letters 69 (1980) 14. [5] J. Wessel, Ph.D. Thesis, University of Chicago. USA (1968); J. Jortner, S.A. Rice and R.M. Hachstrasser, in: Advances in photochemistry, eds. B.G. Pitts and G. Hammond New York, 1969). C63 R.L. F&on and M. Gouterman, J. Chem. Phys. 35 (1961) 1059; 41 (1964) 2280. 171 D.S. McClure,Can.J. Chem.36 (1958) 59. [83 J. Brickmann. Chem. Phys. 24 (1977) 367. [9] S.F. Fischer and N. Heider, in preparation. [IO] L.S. Cederbaum, W. Domcke, H. Kiippel and W. van Niessen, Chem. Phys. 26 (1977) 169.

(WiIey,