Discrete-state approach to the time evolution of molecular states

Chemical Physics 139 (1989) 265-281 Non-Homed

DISCRETE-STATE APPROACH TO THE TIME EVOLUTION OF MOLECULAR STATES Giuseppe DEL RE ‘, Wolfgang FORNER, Detlef ~G~ANN

and J&IOSLADIK

Chairfor TheoreticalChemistry,Friedrich-Alexander-University Erlangen-Niirnberg, D-8520 Erlangen, Egerlandstrasse3, FRG Received 14 February 1989; in final form 4 August 1989

The generalfo~~isrn for dete~ining Born pm~b~ti~ associated with transitions between sets of coupled localized states of molecules is developed and discussed. Special emphasis has been put to the nature of the states involved and to the filtering effects of preparation and detection in processes like redox action involving distant sites of the same redox molecule. Numerical examples are given, and steps to be followed in the analysis of a concrete case are illustrated in detail on preliminary results. These were obtained in studying intramolecular electron transfer in the iron-imidaxole part of a heme group.

1. Introduction Several research groups have tackled the problem of adapting the theory of the time evolution of quantum mechanical states to molecular problems, and many general conceptual questions, such as the preparation of initial states, interference effects, etc., have been raised and substantially settled. In addition to classical treatment the general quantum mechanical framework of the subject, like those of Goldberger and Watson [ 1 ] and of Levine [ 21, specific studies were published by Robinson and his school [ 31, and by Jortner, Freed, and their co-workers [ 4-71. Phenomena such as intramole~ul~ radiationless transitions [ 8 1, electron transfer [ 91, and energy transfer [ lo] have been discussed in the light of those studies. More recently, Lami and Villani [ 11,12 ] have treated applications to situations that can be studied experimentally with lasers. The published analyses are mostly based on representations of dense state manifolds in the form of continua, so as to allow applications of the Fermi golden rule and similar expressions. This is appropriate for general analysis, because of the large density of vibrational states in large molecules. Nevertheless, the alternative point of view - coupling of a few sparse Permanentaddress:CattedraChimicaTeorica,Universitadi Napoli,80134 Napoli, via Mexxocannone 4, Italy. 0301-01~/89/$03.50 (North-Holland )

0 Elsevier Science Publishers B.V.

states - deserves attention when the couplings and separations of a few specific states are expected to be much more important than every other coupling or separation and much different from one another, for in that case the phenomena under study are essentially controlled by the time evolution of those states. Considerations of such a “discrete-state scheme” is also interesting because large discrete sets of states can be handled by computers, and thus the way is open to the exploration of detailed decay mechanisms requiring a choice between competing channels involving several quantum levels with greatly different couplings and separations. In fact, in spite of solvents and other perturbations, many phenomena of molecular science can be explained by models involving only few quantum states. Two-state systems are known to play a central role, e.g. in the theory of NMR and ESR. In dynamical intramolecular problems treated in terms of quantum transitions and time-dependent probabilities, there are two or three relevant partners, e.g. the macrocycle, iron, and imidazole in heme groups [ I 3 1, or the ruthenium units in trinuclear Ru complexes [ 141 in the case of intramolecular electron transfer [ 1S], etc. This suggests that the relevant states may be a few members of discrete state sets; it seems reasonable to expect that coupling to the continua pertaining to the solvent states or to the dissociated states will cause delay [ 16,111 and losses in efficiency, but

266

G. Del Re et al. / Time evolution of molecular states

will not control the main steps of the mechanism. As a contribution towards making easier applications of the quantum mechanical theory to problems of the above type, we shall discuss in this paper the time evolution of states belonging to coupled discrete sets. The treatment given may be seen to some extent as a generalization and an extension of work by Hall and Nobutoki [ 17 1. Consider first of all the case of a two-state system. The probability of transition from state I 1) to state 12) is W,2=(sin~sinot)2,

(1)

where

The time-dependent counterpart of this basis (viz. the set of states at time t which were those of eq. ( 3 ) at time 0) may be written as

(4) In the case of orthonormal states the Born probability of finding 1j) in I k( t) ) is wkj=l



I*.

If the states under consideration do not form an orthonormal set, eq. ( 5 ) may be generalized to wjk=l

(iIlk(

If

={
H,,

-H2*

’

(2)

with Hjk an element of the appropriate Hamiltonian matrix. The period is 1 ps when fio~209.44 cm-‘, slightly less than 1 mhartree. This is the order of magnitude we must expect for the coupling between degenerate states, in the case of a physically observable transition. Coupling up to two orders of magnitude larger will still give transition times comparable with vibrational relaxation times. Eqs. ( 1) and (2) also remind that the condition H, , - HZ2 x 0 is critical if the transition probability must be reasonably high (energy conservation ) .

2. Quantum formalism In a discrete-state approach it is possible to use standard matrix algebra without continuous matrices. This is important for computer applications. We want the probability that a state lj) be found at time t by observation on a system that had been prepared in state Ik) at time 0, and whose state at time t is thereforecalled )k(t)) (lk(O))=lk)).Sinceboth the initial and the final state are admissible states of the system (in the sense of quantum mechanics), they can be taken, together with all other states that can be observed (or prepared) by the same type of experiment, as a time-independent basis 1~)=(11)12)...lj)...lk)...).

(3)

(5)

(5’)

where the bar denotes an element of the dual state space and subscript s denotes that the symmetrized form has to be used:

Otherwise e.g. for the projector $j (which is in symmetrized form given by $ ( IT) (jl + Ij) (;I ) ), bj=fij’ would not hold. We shall work with eq. ( 5 0 ), but we warn the reader that this more general case involves a number of conceptual and numerical difficulties, as explained in appendix A. Let W” denote a diagonal matrix whose (k, k)th element is the probability (e.g. the Boltzmann probability) that the given system is in state I k) at t =O. Let P be the diagonal projection matrix whose (j, j)th element is 1 if Ij) is one of the final states of interest, 0 otherwise. Then, using the letter K to refer to the set of initial states and the letter J to refer to the set of final states, the total probability that a transition from any state of the initial subset to any state of the final subset has taken place at time t is given by w,=F~P”I(“k(‘))IfW~k.

(6)

This expression, after suitable transformations, will yield our master equation. The explicit time dependence of I k(t) ) is given by the Schriidinger equation

261

G. Del Re et al. / Time evolutionof molecularstates

(Aj#)=(~&f>))

A”(t)=A(t)S-’

(7) Multiplication of eq. (7) on the left by If> and use of the property

and A”(t)=SA(t)S-’

(A$(t)=(j/&))).

Now eq. (6) can be written as If> (xl = Ix> (fl =I > IX) = Ix)S-’

(8)

(where S= (x1x> and 4 represents the unit operator) gives ifi$
(9)

.

Let Ak denote the “amplitude vector” (Xl k( t) > . Then eq. (9 ) becomes

+A~A~+A~A~)

+AjkAT W&

[P(A~A~‘++A’~A~~+

= -$ I

+A”‘W”A++A”~A’+)], =t Tr[P(AWOA’“++A’WA”+

ifi&Bk=H’Bk,

(11)

(W” and P are diagonal matrices). Substitution of eq. ( 16) into eq. (6) and use of the cyclic invariance property of traces yields

Wm=Tr[exp(-iiwt)

M exp(iwt) NJ ,

3

B,=S’/zA

1/2WOS1/2)T,

N=: ~T+(S”2PS-“2+S-“ZPS”2)T.

Note that for S= 1 eq. ( 19 ) reduces to M =T+W”T and N =T+PT.

Let T be a unitary matrix such that

ho=T’H’T

(13)

is a diagonal matrix. Then, as is well known, eq. ( 11) gives Bk(t)=TeXp(-iot)T+B,(O),

(14)

and, because of eq. ( 12 ) ,

From eqs. ( 16 ) and ( 8 ) we obtain b$ck(t)=(.AQt))),

Eq. ( 18 ) can be further transfo~ed by replacing the complex exponential by its t~gonomet~c expression (note that the imaginary part vanishes because the probability W, is real ) . The result is Wm=Tr R(t),

(20)

where .

(15)

If the system is initially in one of the basis states, the appropriate value of A&(O) is the Kronecker symbol. Therefore, the whole matrix A(0) is the unit matrix, and eq. ( 15) becomes A(t) zS-~‘~T exp( -iot)T+S”2.

(19)

(12)

k.

exp( -iot)T+S”2Ak(0)

(18)

where M=:fT+(S”2wS-“2+S-

where H’__S-1/2HS-l/2

(17)

(10)

(Ak is the kth column of a matrix A, H is (x{ &I 2). f Using Liiwdin’s classical transfo~ation [ 18 1, we multiply eq. ( 10) by S’12 and obtain

A’(t)==(t)

$ C Pa(AjkAr P

+A”‘W,A++A”W”A’+)]

ih %A*=S-‘HA*.

t&(t) =s-‘I?

Ww=

R(t)=cos(wt)M +sin(wt)M

cos(ot)N sin(wt)N .

(21)

If the argument of the trace is written in explicit summation form, eq. (2 1) gives L(t)=

c [cos(o,,t)Iw,,cos(w,t)N, ”

(16)

+sin(o,,t)M,,sin(o,t)N,] =CM,,N,cosI(w,,-o,)tl* Y

(22)

268

G. Del Re et al. / Time evolution of molecular states

Finally, summation over u yields the trace of eq. (20) in the form of a Fourier sum (23) U,”

with Q”v=d~,-%,,

&,(0)=&N,.

(24)

This is our master equation.

3. Preparation and detection of the relevant states. Elimination of high-frequency components The processes to which the above treatment can be applied are, in addition to special cases of the wellknown intramolecular relaxation processes, those relaxation processes which involve very large systems consisting of well-defined entities, such as molecular crystals, certain polymers, and biologically active macromolecules. Initial states can, of course, be prepared by laser light or by other techniques chosen by the observer. Such things have been widely discussed, and we refer the reader in particular to refs. [ 6,11,19]. The question is less clear when a process such as electron transfer is to be described. As far as we know, treatments of this class of processes [ 9,201 normally assume some reasonable initial states without pausing on the mechanism by which they are prepared. Given the complexity of the systems to be treated, little more can be done unless mechanisms are specified. An example of what this involves is presented in section 5. In general, it may be expected that the initial state is determined by approach and detachment - or by another form of activation such as radiative excitation - of a molecular species D (e.g. an electron donor) not included in the process to be studied, and hence treated as a “preparation device”. Therefore, the initial states are those in which the system under study is left after D has performed its task. For example (cf. section 5), in a photosynthetic centre, the light absorbing region will yield an electron to a cytochrome chain starting with a porphyrin macrocycle. The latter will thus be left in a localized negative-ion state, which is not a stationary state of the cytochrome chain under consideration. The transfer process along the chain will thus be initiated.

The same consideration holds for the final states, which are those states which enable the “detection” state of the process to take place, for instance the reception of an electron by an electron acceptor A which will react in some specific way (as happens in the reduction of CO* in chloroplasts). The species A too must then be treated as a system different from the one under study: in fact, it plays the role of a measuring probe. In connection with preparation and detection, it is important to take into account aspects of the interaction of D and A with the system under consideration that are not especially important at a purely abstract level. In particular, it is necessary that the time required for the measurement and the preparation processes should be included in the computations. We now proceed to a simple analysis of this particular point. Eq. (23 ) represents a theoretical probability of finding the system in a condition common to any one of states of the set J (say, localization on the righthand side of a double-well potential energy profile) if it was originally in a condition common to the states of the set K how far is that probability a description of reality? Are there further transformations to be applied? One condition is of course the validity of the discrete-level picture for the problem at hand. Mathematically, it requires that the continua to which the levels are always coupled should not induce broadenings comparable to the energy separation and/or the direct coupling between the discrete levels under consideration (viz. that the damping term as used in ref. [ 111 should be small). A completely different difficulty arises from the fact that the very notion of a specific property of states of either set implies at least in principle an “observation” made to detect that specific property. For instance in the case of electron transfer, a reaction whose outcome will be different depending on whether the mobile electron is on the donor or on the acceptor. Now, the observation in question takes time, so that it will perform a mean on the ideal probability: high-frequency components will be ignored, whereas low-frequency ones (below a certain threshold) will be detected. The “measuring apparatus” (e.g., the reduced part of a given enzyme or the external reduced molecule) may be considered

G. Del Re et al. / Time evolutionof mokcular states

as an interval 2a. Then the actual probability serving the transition at time t is I+fl W(t)=

$

j W(t’) *-r7

dt’ ,

of ob-

(25)

with W(t’)=C

“”

R,,(O)

cos(S2,,t’)

.

The required time average involves form *+n I,,(t)=

$j

(26) integrals

269

Eq. (30) is the expression to be used to represent direct physical observations. Unfortunately, it is difficult to find estimates for the preparation and detection times. In phenomena like electron transfer, physical intuition suggests that they should be comparable with the classical period of molecular vibrations, viz. between 0.0 1 and 0.1 ps. This places them well below the relaxation times of solvents at ordinary temperatures.

of the 4. Numerical results

cos(SL,,t’) dt’

I-0

=cos(QwJtk(2~~lTu”)

(27)

1

where g(x) = (sin x)/x, T,,=2n/Q,, The wellknown properties of g(x) indicate that the average becomes negligible for CD Tug for CJ= T,,, we have already g=O.O5. Substitution of eq. (27) into expression (25 ) for the average probability gives ~(t)=~R~,cos(sz,“t)) I(,”

(28)

with &=&,(O)

g(2no/T,,)

.

(29)

Expression (28) is not final, because it only takes into account the uncertainty of the exact time at which observation of the final state has taken place. Now, quantum mechanical uncertainty also affects the phase of the observed signal, since also the instant at which the initial state was prepared is not sharply defined. Therefore, it is necessary to take into account that the signal received at time t had started at an unspecified instant between - 6’ and a’; so that, even if the detection had taken place instantaneously, the signal detected at time t would be in the average a superposition of signals originated at different times. Therefore, a further averaging of eq. (28) between t-u’ and t+a’ is necessary. Here we shall assume that the “observation time” is the same as cr. Therefore, our final formula will be W”‘(t) = 1 R; c0s(f2~,t) I.0

,

(30)

I’.

(31)

with KZ =&JO)

[g(2na/T,,)

As is well known, when a discrete-state model is used, all processes are necessarily periodic (Poincare recurrence time), unless the conditions for deterministic chaos are satisfied. In cases such as the inversion of ammonia [ 2 1] the periodicity is observed, and may provide interesting applications. If the detection and measurement processes described in section 3 make the phenomenon under study irreversible - as when reduction is followed by formation of a new molecule - the Poincart recurrence time is of limited interest, and what really matters is some length of time, after which the evolution of probabilities looses its physical significance. In most cases, this “physical” time limit will coincide with the time at which the probability of finding the system in one of the final states has become larger than a certain value, to be estimated in each case by suitable arguments. Such arguments are of the kind used in the general theory of excited-state lifetimes. The point about the physical time limit may concern in practice just the longer periods of the Fourier structure of the Born transition probabilities under consideration, because shorter oscillations may be too fast to produce any effect on the “observing” system. As has been explained in section 3, averaging procedures are then useful to facilitate the interpretation of results in terms of observable phenomena. Also the possibility that the expected “detection” takes place not after the first cycle but after a subsequent one must be taken into account; in this case the temporal evolution of the overall macroscopic phenomenon might present a sort of quantized fine structure. Such theoretical possibilities, already supported by more or less direct experimental evidence such as the ammonia

270

G. llel Re et al. / Time #olu~io~ of rno~~~l~r states

inversion mentioned above, will become the subject of ~xperimentai studies as the techniques for the analysis of very fast processes become more current in chemical studies. This premise should be held in mind in an assessment of the physical significance of the nume~c~ results presented in this section. We also call the attention of the reader to the similarity of certain models with couplings of discrete levels with continua, such as the quantum beats of fig. 5 below. The effect of additional couplings of the discrete-state system taken into explicit consideration with dense manifolds such as solvent states can be treated following the work of Lami and Villani [ 111. As a first example consider the state manifold shown in fig. 1. It consists of a set of eight states I I), .... 18).Thestate 11) atOcm-‘iscoupledby33cm-’ to the state I 5 } , which has the same energy; the seven states 12) to 18> form a sequence with 10 cm-’ step, Bnergy

-1

(cm

)

30

20

‘0

0

-10,

-20.

-30. ----

couplingparameter33.5

~~~~_~~coupling

parameter

-I cm

a

Fig. I. The model level scheme used for the calculations for figs. 2 and 3.

and are coupled to one another by the same energy Q (table 1). A state a~angement of this kind can be used to model a situation where a system formed by a transition metal atom coupled to a ligand is prepared with an excitation on the heavy atom, and the resulting energy transfer to the ligand is studied. The coupling of the seven states localized on the ligand may be seen as a consequence of the perturbation brought about by the heavy atom on the isolated-ring stationary states. Let us assume that the system is initially in state I 1) : it will be enough to study the probability W, , of finding the system still in state 11 } at a later time t. When a=0 (no coupling between the states 12) ... 18) ), a simple oscillation between 1 and 5, with a period of 0.498 ps is found as expected. Fig. 2 shows the results as a is allowed to increase to 20, 60, and 100 cm- r. For a small coupling (fig. 2a) an oscillation with a large amplitude is still present.The Fourier analysis shown in table 2 is dominated by two components with periods close to 0.5 ps and amplitudes close to 0.15 and 0.04. With increasing coupling the dominant amplitudes become smaller and the frequencies higher (shorter periods). As a consequence, the mean value (over reasonable times o, cf. eq. (3) ) of the probability of finding the system in state 11) increases with t, which means that the excitation transfer becomes less efficient. This effect, which bears a close analogy to the stabilization of aromatic rings, may be important in attempts to explain electron and energy transfer across sections of biomolecules [ 11. Its explanation is seen immediately if the “reduced coupling” form of the Hamiltonian matrix is considered. To obtain that form, an orthogonal transformation is applied that diagonalizes the 2-8 block. The new states 12) to I 8) are thus uncoupled to one another, and all coupled to / 1); however, for values of a that are not very small their separation in energy from I I} becomes large (even for state 15> ), and therefore the transition probabilities have short periods and small amplitudes. We can thus speak of a “transfer blocking effect” of systems having dense groups of local states all very sensitive to perturbations. Application of the averaging described in section 3 has been performed according to eq. (28) (table 3). With a= 100 cm-’ and values of CJof 0.01, 0.1 and 1 ps, the results are as shown in figs. 3a-3c. Whereas the lower values of

271

G. Del Re et al. / Time evolution of molecular states

Table 1 Hamiltonian matrix used for the calculation of table 2 and fig. 2 (in cm-‘) where a is varied (corresponding to the level scheme in fig. 1) 1 0

4

3

2 0

0

-30 -2;

0

a a -10

5

6

33.5 a a

0 a a a a 10

:

I

8 0 a a a a

0 a a a a a

2: 3:

u do not greatly affect the time evolution of W, ,, when (Tis 1 ps only the long period components (3.2 and 4.9 ps) survive, modulated by a 0.28 ps component. In the case treated the amplitude of the surviving oscillation is quite small, but it is possible to show that there are more favourable cases. Fig. 4 illustrates another system of levels, which may correspond roughly to the situation found for the keto-enol tautomerism in the systems like the formamide dimer or in systems like those treated in ref. [ 111. Once again we have two degenerate levels, coupled by 100 cm-’ to one another. However, in this model state 11) is coupled weakly (parameter p) to a set of levels about 100 cm-’ above I 1) and separated by only 1 cm-’ from each other. A second set of levels lies about 100 cm- ’ below 12) and is coupled weakly to 12). These two level sets may simulate an environment or, in double-well systems like the formamide dimer, simply a subset of the vibrational modes of the keto and the enol form. The probability W,, of finding the system in state 12) after preparation of I 1) at time 0, with a coupling parameter p of 1 cm-’ is shown in fig. 5a. The most interesting feature is the occurrence of pulses with a period of c 40 ps and a width of zz 10 ps. Between these pulses W,, is almost negligible. If p< 1 cm-‘, the pulses broaden continuously until for p= 0 (two-level system ) , WI2 takes up the typical fast-oscillating behaviour. If p increases, the small peaks between the pulses in fig. Sa become larger, and for p=2 cm-’ (fig. 5b) they dominate the picture. Also the height of the pulses decreases with increasing p. Figs. 5c and 5d show WI*(t) for p= 1 cm-’ with a first-order averaging using a=O. 1 and 1 ps, respectively. For a=O.Ol ps we obtain fig. 5a nearly unchanged. If the

measuring experiment needs 0.1 ps the fast oscillations within the peaks seen in fig. 5a are more or less averaged out, and for g= 1 ps they vanish completely (fig. 5d). The most interesting feature, the pulse structure, occurs only within a very small range of the coupling parameter. Also for sets of less than seven “environment” states the pulse structure disappears. However, there are other parameter sets for which such a structure can be found. For p= 1 cm-’ the spectrum is dominated by two large periods (33.8 and 17.7 ps) with amplitudes of 0.012 and 0.011, and two small ones (0.16 ps both) with amplitudes of 0.0 11. However, there are many other oscillations with amplitudes in the same range of magnitude. The peak height decreases in time due to oscillations of very large period. Fig. 5e, where the results of a long-time simulation for p= 1 cm- ’ with a coarser mesh (0.1 ps instead of 0.01 ps) are reported, shows that the pulse heights decrease with time. However, the structure itself is still present after 1 ns (although, of course, after a longer time, the whole trend will be reproduced, because the whole scheme is periodic).

5. Guidelines for the analysis of specific systems

So far, it has been shown that it is possible to use the discrete state scheme to investigate features of time-dependent phenomena that depend on special arrangements of states. In this way in principle such details of the level distribution can be studied that cannot be taken into account in crude treatments such as applications of Fermi’s golden rule. We now intend to show how the whole formalism can be used

273

G. Del Re et al. / Time evolutionof molecularstates

Table 2 Periods T,j (in ps) and amplitudes R,( 0) (multiplied by + 4) for the largest components obtained from the Hamiltonian shown in table 1 a=20

a=60 cm-*

cm-’

a= lOOcm_’

T@

14&I

i

i

Tfj

l4W

i

j

T#j

14&l

7

0.51

7 7 7 2 3 8 4 7 8 3 4 6 4 8 8 6 8

0.48 0.58 0.75 6.93 4.23 0.34 1.62 1.34 0.20 2.62 1.31 0.83 2.63 0.20 0.21 0.74 0.23

0.593 0.155 0.086 0.063 0.059 0.033 0.03 1 0.024 0.013 0.012 0.009 0.006 0.005 0.004 0.003 0.002 0.001 0.001

3 2 4 1 2 3 6 2 7 1

7 7 7 7 3 4 7 4 8 3

0.39 0.37 0.45 0.33 6.32 2.83 0.60 1.96 0.10 2.27

0.258 0.161 0.083 0.015 0.015 0.008 0.006 0.005 0.004 0.001

3 4 2 1 6 3 2 7

7 7 7 7 7 4 3 8

0.28 0.31 0.27 0.25 0.37 3.22 4.93 0.06

0.163 0.070 0.038 0.005 0.003 0.003 0.002 0.002

i

i

2 1 3 4 1 2 7 2 6 2 1 1 2 3 1 3 1 4

Table 3 Same as table 2 for a= 100 cm-‘, S = 1 and an average for different values of u (period given only in first case, because they remain unchanged) a=1 ps

o=O.l ps

a=O.Ol ps i

j

TfJ

14&l

i

j

14Rzjl

i

i

14&I

3 4 2 1 6 3 2 7

7 7 7 7 7 4 3 8

0.28 0.31 0.27 0.25 0.37 3.22 4.93 0.06

0.162 0.069 0.037 0.005 0.003 0.003 0.002 0.001

3 4 2 3 6 2 1

7 7 7 4 7 3 7

0.067 0.034 0.037 0.003 0.002 0.002 0.002

3 3 2

7 4 3

0.003 0.002 0.001

in concrete, usually complicated problems. Special emphasis must then be placed on the following questions: how can the initial and final states be specified for a given molecule? How many states must be involved in a model suitable of representing a real process? Along what lines can estimates of the energies and couplings of the pertinent states be obtained? These are by no means trivial questions, since the states that must be considered are in general non-stationary vibronic states of large molecules, and a large

amount of computer-assisted guesswork is necessary. We shall explain here the lines along which this kind of analysis is being carried out in work in progress in our laboratories on the time dependence of intramolecular electron transfer in the heme groups of cytochromes. Consider the complex between iron Fe( II ) and hydrogenated imidazole ImH, which may be considered as the relay system of electrons received from the porphyrin macrocycle [ 15 1. Let X+ be the elec-

_ --_

275

G. Del Re et al. / Time evolutionof molecularstates

Energy

to environment effects, the positive hole present in imidazole in 1A) may be filled by an electron localized on Fez+ with an energy expense much lower than the appropriate free-ion ionization potential ( z 30 eV). This transfer would lead to the (possibly longlived ) state

km-‘)

103 101 99

IB) = lFe3+ImH)

.

(33)

97 I

(The vibrational contribution to states I A) and I B > still undefined.) In line with the spirit of the present scheme, we also assume that - at least in biochemical systems such as cytochromes where Y- is the macrocycle having received an electron from another enzyme of the redox chain -the reduction step (Ib) takes place before the imidazole ring has had any chance to give the electron back to iron (as would happen in due course in a discrete state scheme, because all transitions are periodic). In order to assess the probability just discussed that the process represented in scheme (Ia), (Ib) includes a step I A) ... I B) it is necessary (i) to further specify the nature of those two states and (ii) to show that, at least in principle, transfer times between them can be sufftciently long. More details should be discussed and ad hoc experimental information would be needed before a final conclusion may be reached, but this is outside the scope of the present paper. Estimation of energies and couplings is a comparatively hard task even for the less complicated electronic contributions, due to the well-known difftculties with handling iron. We shall base our considerations on minimal basis set calculations on Fe(II)ImH carried out using the program package GAUSSIAN 82 [ 221 in its CDC adapted form #I. The STO-3G basis provided by the program was used for all atoms except iron, for which a comparable minimal basis was used [ 241. Geometry optimization on the singlet state of the system gives the geometry shown in fig. 6. The total energy obtained for this state is - 1478.2 11580 hartree. A computation on the quintuplet state by the unrestricted Hat-tree-Fock (UHF) method would lead to large spin contaminations; therefore, the anis

0

-i

1. .. ..

i

I’>

9;. :::;.. ‘,..>.’ ._y._.:..

‘i:;T).b .. .. .;.. : ,....-. . 5.

-97

.;,. . ....*‘.. .:..,.. : . . .. . . . .. : ., .. . .. . ...._-

-99

..:-,.-._ .:’ .. ., .-

-101

-.:

- 103

_

m-e

. .... ...

coupling

parameter

100

coupling

parameter

p

cm

:.---

-I

i

Fig. 4.

The level scheme leading to pulse structures.

tron acceptor to which imidazole gives up the mobile electron, Y- the electron donor (the macrocycle). The whole process to be studied may be assumed to consist of two steps Fe(II)ImH+X++X+

[Fe(II)ImH]+

[Fe(II)ImH]++Y--+Y+Fe(II)ImH.

,

(Ia) WI

The charged complex in (Ia) and (Ib) may be a completely delocalized system; but it is reasonable to wonder whether, on the contrary, the electron yielded to X+ could be initially localized on the aromatic IC system, so that in (Ia) the charged complex is in a transient state that can be represented as IA)=

IFe’+ImH+)

.

(32)

This can well be a long-lived (quasi-stationary) state, since, as will be discussed in more detail below, the highest occupied canonical orbital of the precursor Fe (II ) ImH is an orbital practically localized on the imidazole ring. The fact that the redox potential Fe’+/Fe’+ is only 0.77 V, as a result of solvent effects, suggests that, due

xl A version of GAUSSIAN 82 adapted to Control Data Corporation computers was provided by T. Kovhi, Computing Center, University Erlangen-Niimberg, FRG. For the implementation of the AUHF method, see ref. [ 231.

2

G. Del Re et al. / Timeevoltition o~~oi~~ar

states

278

Fig. 6. The garnets

6. LIeIRe et al. / Time evolution o~rno~~ulor states

of the Fe(1I)ImI-I system optimized for the singlet state using the restricted Hartree-Fcxk method (planar geometry).

nihilated UHF (AUHF) method developed and implemented into GAUSSIAN 82 by Kovaf [ 23 ] has been applied. In this method in each iteration cycle the next higher contaminating spin component to the desired state is annihilated. An expectation value of s* of 6.00 1 was obtained for the quintuplet state. The latter has turned out to be more stable than the singlet by 5.30 eV, in agreement with previously publishedsemi-empirical calculations ( 6.0 eV [ 25 ] ) . The energy eigenvalues around the highest occupied molecular orbital (HOMO) as well as the characters of the co~es~nding orbitals are shown in table 4. There are at least two orbitals that are localized so as to be meaningful for assessing our electron transfer hypothesis: the n-type lone pair of iron, lying between - 0.9 and - 1.O hartree, and the HOMO (also of n: type) at about -0.6 hat-tree and almost completely localized on the imidazole ring. As has been mentioned, the Fe(II)ImH system may be considered as the precursor of the two states of interest 1A} and I B) . Upon ionization the levels are expected to retain their localization, but will certainly change their energies. The increase in energy of the (ring-x) HOMO can be estimated as the mean

value of the ionization potentials of the three carbon atoms and the two different nitrogen atoms [ 26 ] :

= 97469 cm-’ = 444 mhartree . This value is probably more realistic than the HOMO orbital energy ( x630 mha~ree} suggested by Koopmans’ theorem. Then, roughly speaking, in I A) the ImH+ IForbital will be at about - 190 mhartree and the contribution of the two orbitals under consideration (a~uming fixed electron repulsion ) will be about - I 110 mhartree. A similar argument will give approximately the same value (- 1065 mhartree) for the corresponding energy contribution to the state where an electron is removed from iron, if the ionization potential of the latter is taken close to 13 eV (cf. appendix B). In conclusion, the oxidized ion can be expected to have two nearly degenerate long-lived states corresponding to eqs. ( 32 ) and ( 33 ) . The gap between the two purely electronic states can probably be bridged by vibrations, since a single quantum of vibrations of the heterocycle or of the metal-nitrogen bond will

G. Del Re et al. / Time evolution of molecular states

279

Table 4 Orbital energies 4 (in hartree) for levels around the HOMO a) computed for the singlet state (restricted HF) and quintuplet state (AUHF, see text) for Fe{ 1I)ImH computed within a minimal basis set in the optimized geometry of the singlet state

- 1.0016 -0.9250 -0.9076 -0.8977 -0.8804 -0.8192 -0.7357 -0.6343 -0.3473 -0.1883 -0.1596 -0.0777 + 0.0002

Fe ImH global ImH ImH imH ImH ml FeN global FeN global ImH

- 1.0547 -0.9195 - 0.9074 -0.8955 -0.8788 -0.8068 -0.7247 -0.6335 -0.3419 -0.0876 +0.0017 +0.2204 +0.2310

Fe ImH global ImH ImH global

ImH ImH .) FeN ImH ImH ImH ImH

-0.9225 -0.9191

ImH Fe

-0.8965 -0.8806 -0.8539 -0.7357 - 0.6307 -0.3565 -0.1143 -0.0851 -0.0698 -0.0524

ImH ImH global ImH ImH a’ FeN global Fe FeN global

‘) Highest occupied molecular orbital (HOMO).

provide between 1 and 15 mhartree. As part of our hypothesis, consider the possibility that the imidazole ring will transfer its hole with excitation of a sufficient number of quanta of the ring pulsation at 1156 cm- ’ (6.27 mhartree) [ 271, since this mode is likely to be affected by the Fe-N stretching (in free imidazole it lies at 1142 cm-’ [ 281). As a further tentative assumption, the decay of the vibrational excitation may be expected to take place primarily via the ring mode at 845 cm-‘, the more so as in iron-imidazole complexes this mode is almost degenerate with combined vibrations of the whole complex [ 261. Thus, the following transfer hypothesis appears to be worth further ex~~mental and theoretical investigation: state IA) is directly coupled to state 1B), which involves excitation of about seven quanta of ring pulsation; state 1B ) is coupled to a number of ring vibrational states where the nitrogen linked to iron is involved, and these provide the fastest relaxation channel. The electronic coupling between I A) and 1B) is certainly weak, since they are orthogonal in the precursor complex, but is not zero because of the electric field due to ionization and because of geometrical distortions. The pertinent value can be estimated using the MNDO recipe [ 291, which consists in treating the states as if they were orthogonal, but using overlap values as pro~~ionality constants yielding coupling elements from the mean state energies. In

our case, we can estimate the Fe-N electronic overlap (from Stater orbitals) to be about 0.2. The vibrational overlap (Franck-Condon factor) can be estimated by assuming that the coupled vibrations are the zero-point Fe-N bond stretching at 211 cm-’ [ 271 and the eighth-order ring pulsation, and computing the overlap of the appropriate Hermite polynomials centered in the neighbourhood of the midpoint of the Fe-N bond and at the midpoint between the N atom and the center of the ring. This gives FC factors of 1.5 x i 0T3 between the primary states, and a term of 0.011 mhartree for the coupling to the relaxation channel suggests a transfer probability of 0.2 after 0.11 ps. These are just indicative values, but they show that further work in this direction may lead to interesting results.

6. Conclusion The general examples and the concrete illustration given above show how the general formalism presented here can be used for estimating times and efficiencies of time-dependent processes at the molecular level. A formidable amount of calculations and a large number of intuitive estimates are necessary to completely solve problems of real interest to chemists. Neve~heless, since the role of theory often consists in trying to determine possible mechanisms of

280

G. Del Re et at. I Time evolulion of mdecuiar states

elementary processes and in analyzing them far enough to make possible the planning of ad hoc experiments, it seems to us that much progress in understanding chemical reality should be possible along the lines presented here. In particular, the discretestate approach represents a fundamental tool for advances in the comparatively new field of time-dependent quantum chemistry.

Appendix A The possibility that the non-stationary states under consideration may form a non-orthogonal set is worth a few comments. Since preparation and detection of quantum states are two independent operations, it is not surprising that initial states may not be orthogonal to the final ones. It is less evident that the initial or final states may form two internally nono~hogon~ sets. When the states are prepared and observed by the same type of experiment, it seems obvious that they should be eigenstates of the same operator, and hence orthogonal. We point out, however, that preparation and observation may be seen as the application of a perturbation (say, an electric field) in which the stationary states of the system do not coincide with those of the isolated system; if the perturbation is suddenly switched off, the states are no longer stationary and start to evolve. With this picture, the prepared states do form an o~hogonal set. However, the implied “sudden approximation” [ 301 may be replaced by an approximation where the states are allowed to relax as part of the preparation, say, the equilibrium geometry is changed. In this sense it is not surprising that they may not be orthogonal. A similar argument can be used for observed states. The approximations involved in the modelling of the whole process may also be responsible for non-orthogonality. A comparitively simple example is the treatment of proton transfer in a on~imensional double well where the initial and final states are represented by harmonic oscillator wavefunctions. Clearly, this whole question calls for a deeper analysis of the very definition (formal and operational ) of the Born probability in the general case of nonorthogonal states. However, one should keep in mind, that for overlapping states probabilities can no longer be added, and correction terms are needed to avoid

double counting of certain contributions [ 3 I]. With our definition ( 5’ ) this double counting does not occur, as numerical tests, as well as (in special cases) analytical considerations show.

Appendix B The third ionization potential of iron (Fe *+AFe3+ +e- ) equals 30 eV, the second one (Fe” *Fe” + e- ) is 25 eV, and the first one is 8 eV. However, if the Fe(II)/Fe(III) oxidation occurs in an environment, as in the case already mentioned of redox potentials, the potential can go down to 0.77 V [ 32 1. Thus, the value of % 12 eV assumed here is not surprising in view of the fact that the electron removed from iron does not go to infinity but stays at the nearby imidazole ring. The distance between iron and the middle of the ring is x; 3 A. The large coefficients in its x-HOMO are at the three carbon atoms. Thus, a crude estimate of the ionization potential in our case can be obtained by calculation of Ferr+...Hb+ using the same minimal basis as above but scaled such that a=2, b= 1. Unrestricted Hartree-Fock computations with a scale factor of 3.0 for the hydrogen 1s function predict 24.000 electrons on iron and thus in fact a point charge at 3 8, distance from Fe*+. The computed total energy is - 1255.850537 versus - 1256.200993 hartree for free Fe’+. There is thus an increase in energy of 0.35 hartree= +9.5 eV. The same calculation gives an energy of 1255.114145 hartree for Fe3’ in the sextuplet state. Therefore, the ionization potential of Fe’+ is decreased from 29.6 eV (note the agreement with experiment) to 20.0 eV. If the point charge is shifted only by 0.5 A closer to Fez+, the potential is further reduced to 18.2 eV. Finally, a simple electrostatic estimate of the interaction energy of five negative point charges with Fe2’ and Fe3+ (distances to iron 1.828 A, charge -0.49 eV as for N in Fe(II}ImH) simulating an octahed~l environment together with ImH, results in a further reduction of roughly 5 eV of the ionization potential of Fe(H).

G. Del Re et al. / Time evolution of molecular states

Acknowledgement We thank Mr. Carlo Adamo (Chair for Theoretical Chemistry, University of Naples) for his help in program testing, in the assignment of vibrational frequencies and for comparison with previously published data. Further we thank Dr. A. Lami (Instituto dei Chimica Quantistica ed Energetica Moleculare de1 CNR, Pisa, Italy) for reading the manuscript and for valuable suggestions. One of us (GDR) acknowledges the grant of a Visiting Professorship by the “Deutsche Forschungsgemeinschaft” (DFG ) , and grants from the Italian CNR and MPI which made possible the preparation of the source computer programs used in this work and in preliminary test studies.

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