Chemical Physics 32 (1978) 143-152 0 North-Holland Publishing Company

ANALYTICAL DETERMINATION OF NON-BORN-OPPENHEIMER MATRIX ELEMENTS

BETWEEN DIFFERENT MOLECULAR ELECTRONIC STATES

Jiirgen BEUCKMANN and Herbert BITT0 UniversitZt Konstanz, Fachbereich Cbemie. O-7750 Konstanz. West Germany

Received 22 December 1977

Matrix elements for the nonadiabatic correction between different adiabatic Born-Oppenheimer (ABO) states are dis- cussed in terms of agenerating function (nonadiabatic coupling function, NAF) which was introduced earlier. Recursion relations are derived for the case where different vibronic states are coupled via a single non-totally symmetric mode. The formalism is applied to the estimation of non-Born-Oppenheimer coupling between vibronic states of pyrazine.

1. Introduction

If the energy separation between electronic states is small, neither crude (CBO), nor adiabatic Bom-Oppen- heimer (ABO) states constitute a good description of the vibronic eigenstates of molecular systems. Nonadiabatic corrections, due to the breakdown of the ABO approximation can no longer be neglected [2]. These couplings between different ABO vibronic states are important in the theory of radiationless decay [3-l I] , vibronic inten- sity borrowing 112-241, and line broadening [5,6,25,26]. The intramolecular coupling matrix elements Tiv kp

’ between the ABO vibronic states ~i(f,Q)e,(Q) and &(r,Q)t&,(Q) are of the general form [4,8] .

Here r are the electronic coordinates, while Q = (Q1, ___,Qf) are the generalized vibrational normal coordinates

Q, = (&fj”o)L’2Q;, (2)

where Mi is the reduced mass in atomic mass units of the ith mode and Qi is measured in atomic length units (e = l,* = 1). The molecular electronic energy (in G$ units) is U(r, Q). Some simplifying assumptions are generally made to analyze the properties of the nonadiabatic coupling matrix (NACM) (TjV,). Thus, the second term in eq. (1) is conventionally neglected [8,27] and the denominator ~~(8) - ek(Q) is ksumed to be a slowly varying function of Q [9,12,27] which can be taken as constant [9,12] _ Nitzau and Jortner [S] presented a perturba- tional scheme to take the Q-dependence of the electronic wavefunctions into account. These authors used a crude approximation of the denominator ~j - ek. More refined models were used by numerous authors [5,9,12,13,27], and some of these models are based on a single non-totally symmetric coupling mode [ 1,929] . It has been point- ed out [l] that the NACM can be reduced in the latter case to the determination of vibrational matrix elements of a generating function A(Q) designated as the nonadiabatic coupling function, NAF. In this model also, only a single non-totally symmetric mode is mcluded, but neither the second term of eq. (1) nor the Q-dependence of ei - ck is neglected [l] .

144 J. Brickmann, H. BittofNon-Born-Oppenheimer matrix elements

In this paper recursional relations for qV kP are derived for a mode! situation where the adiabatic potentials ei(Q) and ek(Q) for the nuclear motion aloig a single non-totally symmetric coupling mode can be approximated by harmonic oscillator potentials [1,22] _

2. Matrix elements of nonadiabatic vibronic coupling

The two Born-Oppenheimer electronic states !$ and S, considered here may be coupled both statically and dynamically via a non-totally symmetric promoting mode; the corresponding coordinate will henceforth be called Q. The adiabatic Born-Oppenheimer (ABO) vibrational potentials Ei(Q), and ek(Q) can be obtained from crude Born-Oppenheimer (CBO) potentials Vi(Q) and &(Q) by taking a static coupling term Ul(Q) into account which transforms antisymmetrically with respect to a reflection at the origin of Q [1,5,20,29]. In the weak coupling hm- it, where the two interacting electronic states Si and Sk are energetically well separated (relative to a typical Herz- berg-Teller coupling term), the CBO and ABO potentials can be expressed in the harmonic approximation with the frequencies Wi, Wk and pi, C?#k, respectively [22]. The static Heraberg-Teller (HT) coupling term Uik(Q) may be linear in Q [1,20,29], i.e.,

Uik = UQ. (3)

In conventional HT_theory dynamic interaction between different electronic states is neglected [30] _ It has re- cently been shown [l] that the matrix elements

%n,?z = (m, ~r-l($i~~l~k)l~. fik’, (4)

between the ABO vibronic states due to dynamic coupling are given in terms of a generating function A(Q) (non- adiabatic coupling function, NAF) as

T,, =i-2$qmrn h ’

= (L$/2L!u)1’2 [(m+l)l’%?r+l, fij]Alrr, Gk’ - ml’*(?n-l.~~lAltz, Qk)]

f (nk/2n,)‘/2 [,I’* ~m,QilAln-l,52k~-(~t1)1’*0n,~ilAln+l,SLk~], (5)

when the vibrational states loin) are replaced by [n, “i’, i.e. the vibrational wavefunction with frequency Qi and quantum number R In eq. (4) and (5) L.) designate an integration over the vibrational coordinate Q while (.,.) represents an integration over the electronic coordinates of the system. The energy is measured in units of a com- mon basic frequency fir-, and generalized length units are used [ 1,231. The NAF is then [l]

with the energy gap

S20A = ~~(0) - ~~(0) = Uk(0) - q(O).

The nonadiabatic coupling terms T,, [eq. (5)] may then be reduced to integrals of the type

I mn = <m,~&4&* + A)/[(AQ* - A)* +BQ* J In, S2k’,

which will be evaluated further.

(7)

(8)

J. Brickmann. H_. Bitto&on-Born-Oppenheimer matrix elements 195

3. Recursion formulas for NAF integrals

To evaluate the matrix elements I,, of the NAF we proceed in a similar manner as for calculating the matrix elements of the lorentzian term [3 l] . Since Q commutes with each elementary function of Q, i.e.

fQ, A(Q)1 = 0, (9) and

by utilizing the spectral decomposition of the identity operation. We now introduce two kinds of creation and annilation operators, corresponding to two different oscillators with frequencies pi and !+, respectively [32].

Here P is the momentum operator in generalized units. From eq. (11) one obtains

(m, SZiIQlv, ai> = (SZ,/2G!,)1~2(m, !$l(Z? Z?)C, !+ = (!C20/2S2$‘2 [(mi-1)‘~26,+1 , u + m”2S~_1,V] ,

and an analogous expression for (m. S2klQlu, C@. We can now evaluate the summation in eq. (10) as

[&&ml ~i]h(Q)]n-l,~~) + (n+1)1/2(m, ailA(Q)ln+l, ak)] (fii/ak)‘12

= (m+l)“‘(m+l, ~i;lilA(Q)ln. Cl,> + m1/2(m- 1, S&lA(Q)ln, sZ,>.

With the deftition 01~ = C&/C& and the notation of eq. (S), one obtains

Cl$zl/2I, + (n+l)ll2Z , n_l m,n+ll - (m+lY’2&+l,n - m1/21,_l,n = 0. By replacing (mi-1) by m in eq. (14), the fural recursion relation becomes

Z m,n =aL[(n+l)/m] “2zm_L u+l ‘(n/m)“2z~_l~_ll - [(m’-l)/ml 1’2zm_l n’ 2 ,

(11)

(12)

(13)

(14)

(15)

For negative values of m or n the integrals Z,, are replaced by zero. Moreover, since A(Q) is an even function of

Qa Tn., vanishes if m + n is an odd integer. All the integrals of eq. (15) can be successively determined from the fust row matrix elements

‘0, =(g-J2 S dQ A(QWJ(~klfiO) 112 Q> exp - ai+ak,2), (16)

by.means of&z hermitian ~~ynomialsH,(5). With

2510 th e new integration variables q = (G!k/C?,-,)‘/2Q, eq. (16) be-

comes

(17)

= -2v(Aq2 + A)[(As2 - A)2 + 4(n,/a&2q2] -I, (18)

146 L Brfc~n, H. B&o/Non-Born-Oppenheimer matrix elements

where A = (a; - w$/(2Q&,)_ One now can distinguish between two cases (i) A = 0, i.e. Wi = c+; the NAF reduces to the lorentzian

with 2 = (A/~v)~!+& > 0. The first row integrals can then be evaluated analytically by the procedure of Gijze- man [27] which leads to a complicated initial integral, or by use of a second recursional relation (see appendix A).

112 q(z). cm

(ii) A f 0, i-e_ wi # wk; the generalized NAF, after partial separation, becomes

X4) = A,&?) + $(4),

where

A$) = C,l(? +Zr), r=o,P,

and the coefficients are

(21)

’ (22)

c, =

2, = 2v*Q,( l-cr)/(A2s2,) - A/A, ZP = 2v2Q2,( l+o)/(A2s2,) - A/A,

u = +[l - AS2,A/(v2s2,)] lp. (23)

The parameters Z, and ZP are either both positive numbers or complex conjugate numbers. (a) If o is a rest quantity, i.e., v2S&-, ~~~~A then x,(q) and X&T) are again lorentzians, and we can proceed

as in case (i) and

Ian =C&(ZJ *C&,(Zp)- (24)

(b) If u is imaginary, i.e., v2&20 <A!&A the generalized NAF becomes

X((z) =D/($ +z) +D*/(*2 +z*), (25)

where

(26) Z =q + ib = (2v2C+,/A2sl, - A/A) - i(2v2s2,1A2Qk)lol.

The first row matrix elements of the coupling term are, in terms of the integralsF, developed in appendix B.

Zen = (X+/C+, 2nn!7r)1’2 Fn. (27)

The recursion relations presented in this section will now be used to calculate matrix elements of the nonadiabatic coupling term [eq. (5)] for pyrazine.

4. Nonadiabatic coupling in pyrazine

Vibronic coupling via a non-totally symmetric out-of-plane hydrogen bending mode has been intensively stud- ied both experimentally [18,21,33-351 and theoretically [13,17,18,20,21,24,33] for pyrazine.

J. Brickmarm. H. Bitto/!!on-Born-Oppenheimer matrix elements 147

Table 1 J!yrazine data a)

Quantity Vapor Benzene host

Wl PO1 b) 0.744 0.810 w2 No1 b) 0.698 0.640 ~2~ c) [sL,] b) 0.431 0.462 SX~ d) [SZ, J b) 0.960 0.987 ” [nolGU] e) 1.2537 1.4009 A [no] b, 7.000 8.000

a) Calculated from observed quantities [24]. b) Measured in units of s~o = 1000 CXI’~. c) Mean value of the differences of the four lowest observed vibrational levels. d) Calculated from HT-NE0 scheme [24 J _ e) Generalized length units as in ref. [23]_

It has been shown in a previous paper [24] that the band positions and intensities of the So-S1 absorption and emission spectra of pyrazine-do and -d4 in vapor phase [21] and in benzene single crystals [18] can be well interpreted in terms of a diabatic Herzberg-Teller non-Born-Oppenheimer (HT.NBO) scheme [14,2 l-23,35]. By a systematic variation of the CBO v&rational frequencies w1 and w2 pertaining to the 10a mode in the cou- pled S1 and Sa electronic states, the energy gap A between these states, and the vibronic coupling coefficient u, it was possible to reproduce ali the observed data within experimental error [24]. Moreover, values of some pa- rarneters such as the adiabatic vibrational frequency a2 of the 10a mode in the S2 state, which has as yet not been observed, was obtained. These data are summarized in table 1. As is to be seen from spectroscopic data [18, 11,33-351, the vibrational motion in the S1 state of pyrazine along the non-totally symmetric coordinate is not harmonic, while ia the S2 state this motion can be well approffimated by a harmonic oscillator [24] . We replaced the anharmonic motion in &e S1 state by a harmonic oscillator wi’J1 an effective frequency S21, which was ob- tained as the mean value of the differences of the four lowest observed vibrational levels. For small vibrational quantum numbers the wavefunctions of the anharmonic oscillator do not substantially differ from those of the effective harmonic oscillator, so that we can use the above described formalism to estimate the nonadiabatic coupling terms.

For both the vapor state and the solid solution, the recursion relations of case (iia) have to be utilized to ob- tain the nonadiabatic coupling matrix element T,, of eq. (3). WithA = 0.03455 for pyrazine vapor (0.12487 for pyrazine in benzene host) and u = 0.923 (O-705), the NAF parameters (ofeq. (27)) are C, = -3.026 (-3_410), CP =-69X1 (--19.027), and& =8.296 (11.058),ZP = 5073.41(463.86) in terms of generalized units (GU) using sLo = 1000 cm-l. The nonadiabatic coupling matrix (NACM) for the S,-S2 coupling in pyrazine vapor is given in table 2 for vibrational quantum numbers n, m < 20. Although the parameters for pyrazine in benzene host differ substantially from the corresponding values in the vapor phase, there is no significant difference in the NACM for the two cases.

From table 2 it is evident that the absolute values of the matrix elements can exceed 500 cm-’ which is ap- proximately half the value (1027 cm-l) found by Chappell and Ross [36] and one of us [23] for static coupling case. However, these relatively large NACM elements do not contribute very much to the line positions of the vibronic bands for pyrazine as obtained by a simple perturbational treatment. The level shifts n~i, due to non- adiabatic coupling between two vibronic levels belonging to different electronic states up to second order per- turbation theory is

i, k= 1.2. (28)

Tab

le 2

N

onad

iabo

tic co

upl

ing m

atri

x ele

men

ts T,m

,2n

for

the S

1 -tip

in

tera

ctio

n in

pyr

azin

e vap

or (i

n w

ave n

um

bers

, cm-’

1

~1~

0123

\

45

67

8 9

10

11

12

13

14

15

16

17

18

19

0 -2

37

-57

1 31

7 -2

22

2 44

1 -1

51

: -1

78

491

-315

48

0 s

93

-429

6

192

-506

7

-48

299

8 -,

lo9

402

9 25

-1

90

10

61

-281

11

-1

3 11

4 12

-3

4 18

2 13

7

-67

14

19

-114

15

-4

39

16

-1

1 70

17

2

-23

18

6 -4

2 .1

9 -1

13

-19

-6

-87

-38

-98

-60

I88

-54

31

-59

419

106

322

157

-540

20

7 -5

27

88

488

-473

54

4 -3

84

-374

56

6 -4

59

5.50

26

1 -5

27

346

-570

-1

73

428

-243

50

1 11

1 -3

20

163

-398

-7

0 22

6

-2

-1

-14

-6

-24

-11

-64

-34

-65

-42

-19

-57

26

-38

175

68

166

106

-19

132

-104

81

-2

74

-158

-1

55

-180

49

7 -4

1 41

4 56

-5

83

310

-562

19

5 55

7 -5

10

590

-430

-4

12

594

-2

-1

-4

-2

-1

-16

-8

-3

-2

-23

-12

-6

-6

-46

-30

-17

-9

-45

-35

-22

-13

-13

-37

-36

-26

16

-23

-34

-29

120

45

-5

-27

122

69

15

-16

22

107

84

36

-36

79

89

54

-171

-8

4 42

83

-1

37

-116

1

66

128

-84

-129

-3

8 17

0 -2

4 -1

22

-69

81

181

35

-98

-21

162

84

-62

-329

-1

01

122

118

J. Brickmnmz. H. Birrof~on-Born-Oppenheimer matrix elements 149

Table 3 Level shifts AE,-~ due to nonadiabatic coupling between the S1 and 52 state of pyre&e in the gas phase (in cm-l)

i\” 0 1 2 3 4 5

1 -8 -23 -30 -36 -44 -50 2 23 51 8% 11% 143 156

If we approximate elm and ezn by the harmonic values elm = !$(m f 4) and e2, = $(n + 4) t A, we obtain

Aq, = 5 1 7-l,,2m12/ [al n-5222m-A+;(S221-~2)],

Aezn = c l~2n,lm12/[~2n - O,m + A + $(a2 - $)] _ m

(29)

The nonadiabatic level shifts A~i, for pyrazine vapor are listed in table 3. Although the values Of A~i, increase up to a magnitude of 100 cm-l, spectroscopic data like vibrational frequencies as calculated from the energy differ- ence of adjacent vibronic levels are only weakly changed. The adiabatic frequency SL, for the vibration of the non-totally symmetric mode is lowered by 8 cm-l while 0, is raised by an approximate value of 26 cm-l, as is

to be seen from the level shifts listed in table 3. Thus it is predicted that the nonadiabatic SI-S2 coupling does not, in this case, contribute substantially to

the line positions of the vibronic bands. This result is in agreement with an earlier conclusion by one of us on the basis of extensive variational calculations [24] _ Larger level shifts or inhomogeneous line broadening are to be expected if-elm and e2,, are accidentally or bearly degenerate, i.e., if

(30)

‘Although the harmonic approximation for cl(Q) and e2(Q) is v$ry crude for pyrazine, it seems unlikely that such degeneracies exist.

Large level shifts and, consequently, the breakdown of a linear perturbation approximation will occur ifAe$, or if the two CBO potentials cl(Q) and Q(Q) cross. In the latter case the NACM elements have to be calculated according to the recursion relations of appendix B.

In summary, we have developed recursion formulas for the calculation of non-Born-Oppenheimer matrix ele- ments between different molecular electronic states within a nonadiabatic vibronic coupling model. In this model it was assumed that the adiabatic energy surfaces can be approximated by distorted oscillator potentials as func- tions of a single non-totally symmetric mode.

This approach was applied to pyraziue, for which we found that the nonadiabatic coupling matrix elements can exceed half the value of the static coupling terms but do not contribute very much to the line position.

Acknowledgement

We like to thank Karl Weiss for helpful discussions and reading the manuscript. The numerical calculations were carried out at the TR 440 computer of the University Konstanz.

Appendix A: Recursion reMion for Kn

The integral

150 1. Brickniam~, II. Bitto/iVon-Born-Oppenheimer matrix elements

K, = s e-p’42H,(4, dq ; Z+q2

(A-1) -m

can be evaluated in analogy to the method used by Rao [31] _ With the series definition of the hermitian polyno- mials

H,(~) = (2# n@$) (2#-’ +n(tz-1) (;T2) (n-3) (2#-4 _ ... . (A.21

K, can be reduced to the integrals

(A.31

where n has to be even. For odd integers n, L, vanishes. The initial vahte of the recursion relation (A.3) can be calculated analytically as

dq = -?- ePZZ erfc (sz1j2) &d/2

> (A-4)

where erfc (x) = 1 - erf (x). From eqs. (A-I)-(A-3) one fir&y obtains

(AS)

K, = (2’3 _ !i!!$) 2”-2L,_2 +n(n-l) (;;2)@-3) +L,_~ _ + ___ _ (A-6)

For large values of flZ112 9 1, the initial integral can be approximated by the asymptotic expansion [37].

K, =Lo=(~"2/flZ) (A.7)

Appendix B: Recursion relation for F,

The integral

can be evaluated as is done in appendix A, i.e.,

Fn ~2 @-!I n 2 n 1! 2 - R,_2+-m.. 1

where

The complex integrals E,, can be reduced to the starting value E. as in eq. (A-3),

~~ = _zK,_, +D ’ -3 -5 ---(n-3) _ 2n/2-lp”-1

tB.1)

tB.2)

(B-3)

(B-4)

Alternatively, the real and imaginary parts R, and iI,, respectively, of En can be used,

J_~Brickmann, IL Bitto/Nan-Born-Oppenheimer matrix elements

R, = -CZR~_~ - bln_2) +X,, 1, =’ -Wn_2 + bR,_,) + Y,,

with

The initial values of the recursion relations (B.5) which need to be calculated numericalk are

Di(q2+,) - Drb dq.

Cl (qLz)2 t b2

References

[l] J. Brickmann, Chem. Phys. 24 (1977) 367. [2) T. Azumi and K. Matsuzak, Photochem. Photobiol. 25 (1977) 315. [3] S.H. Lin, J. Chem. Phys.44 (1966) 3759. [4] R. Bison and J. Jortner, J. Chem. Phys. 48 (1968) 715; 50 (1969) 4061. [S] B_ Scharf and R. Silbey. Chem. Phys. Letters 9 (1971) 125. [6] B. Scharf, Chem. Phys. Letters 14 (1972) 315. [7] D.F. HeUer, K.F. Freed and W.M. Gelbart, J. Chem. Phys. 56 (1972) 2309. [8] A. Nitzan and J. Jortner, J.Chem. Phys. 56 (1972) 3360. [9] G.C. Marconi and G. Orlandi, Chem. Phys. Letters 21 (1973) 89.

[lo] S.H. Lii, J. Chem. Phys. 58.(1973) 5760. [ll] F. Metz, Elektronische Relaxations aus angeregten Molekiilzust%den, Habilitationsschrift, TU Miinchen (1977). [12] G. Orlandi and W. Siebrand, Chem. Phys. Letters 8 (1971) 473. [13] P.A. Geldof, R.P.H. Rettschnick and G.J. Hoytink, Chem. Phys. Letters 10 (1971) 549. [14] R.L. Fulton, J. Chem. Phys. 56 (1972) 1210. [15] A.R. Gregory and W. Siebrand, Chem. Phys. Letters 29 (1974) 13. 1161 N. Kanamaru and E.C. Lim, Chem. Phys. 10 (1975) 141. [17] N. Kanamaru and E.C. Lim, Chem. Phys. Letters 35 (1975) 303. [18] D.L. Narva and D.S. McClure, Chem. Phys. Letters ll(l975) 151. [19] A.R. Gregory, W. Siebrand and M-2. Zgierski, J. Chem. Phys. 64 (1976) 3145. [20] J. Brickmann, Ber. Bunsenges. Physik. Chem. 80 (1976) 917. [21] G. Fischer, S. Jacobson and B. Scharf, Chem. Phys. 16 (1976) 237.

[22] A.R. Gregory, W.H. Henneker, W. Siebrand and M.Z. Zgierski, J. Chem. Phys. 65 (1976) 2071. [23] J. Brickmann. Mol. Phys. 35 (1978) 155. [24] J. Brickmann. to be published. 2251 B. Scharf and R. Silbey, Chem. Phys. Letters4 (1970) 561. 1261 B. Scharf and R. Silbey, Chem. Phys. Letters4 (1969) 423. [27] 0.L-l. Gijzeman, Chem. Phys. Letters 26 (1974) 152. [28] K.F. Freed and W-M. Gelbart, Chem. Phys. Letters 10 (1971) 187. [29 [3 d

M.Z. Zgierski and A.P. Penner, Chem. Phys. Letters 45 (1977) 41. J J. Jortner, SA. Rice and R.M. Hochstrasser, Advan. Photochem. 7 (1969) 149;

E.W. Schlag, S. Schneider and S.F. Fischer, Ann. Rev. Phys. Chem. 22 (1971) 465- [31] Ch.v.S. Ramachandra Rao, Z. Naturforsch. 31a (1976) 553. [32] J. Katriel, J. Phys. 38 (1970) 1315. [33] G. Fischer and R. Naaman,Chem. Phys. 12 (1976) 367. [34] K.K. Innes, J.P. Simmons and S.G. Tilford, J. Mol. Spectry. 11 (1963) 257;

K.K. Innes, J.P. Byrne and I.G. Ross, J. Mol. Spectry. 22 (1967) 125;

151

(B.5)

(B-6)

(B-7)

152 _ J. Brickmann, I-I. B&o/Non-Born-Oppenheimer mati ekmenrs

S.N. Thaku~ and K.K. innes, J. Mol. Spectry. 52 (1974) 130; I. Suzuka, N. Mikami and M. Ito, J. Mol. Spectry. 52 (1974) 21.

[35] R.L. F&on and M. Gouterman, J. Chem. Phys. 35 (1961) 1059; 41(1964) 2280. * (361 PJ. Chappel and LG. RosqChem. Phys. Letters 43 (1976) 303. [371 M. Abramowitz and LA. Stegun, eds., Handbook of mathematical functions (Dover Publ., New York, 1970) p. 296.