2162
J. Phys. Chem. 1992, 96,2162-2111
Instability versus Blstabillty in Electron-Transfer Problems: The Conflict between Electronic or Nuclear Relaxation and Electronic Delocalization G. Durand,*vf 0. K. Kabbaj,t**M. B. Lepetit; J. P. Malrieu,s and J. Martitqll Laboratoire de Physique Quantique (U.R.A. 505 du C.N.R.S.),UniversitC Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France, and LOE-CEMES, 29 rue J. Marvig, 31055 Toulouse Cedex, France (Received: June 13, 1991; In Final Form: November 5, 1991)
The possible existence of two stable states connected by a single electron transfer has been studied in two different cases: (i) the cation of a molecular dimer (Az)+,which is necessarily symmetrical (with a delocalized hole) at short enough intermolecular distances, and symmetry-broken (with a localized hole) at long intermolecular distances. The symmetry-breaking appears as a bifurcation, as shown through model studies of the (H2-H2)+ dimer. It takes place when the relaxation energy of the localized forms prevails over the left-right hole-delocalization resonance energy. At the Hartree-Fock (HF) level with a rectangular geometry, the conflict takes the form of an instability of the symmetry-adapted solution for intermolecular distances larger than a critical value; the behavior is exactly the same for the exact solutions of the electronic Born-Oppenheimer Hamiltonian with respect to the rectangular trapezoid geometric distortion, so that the H F instability may be looked at as a metaphor (involving electronic relaxation) of a more physical phenomenon involving the nuclear relaxation. (ii) the donoracceptor (DA) complexes and the DA D+A- transition in which the m a t stable structure is ionic at short intermolecular distance and neutral at large intermolecular distance. The problem is analyzed as the matching of two valleys, and two solutions are possible: either an oblique passage from one channel to the other or the coexistence in a certain domain of intersystem distances of two parallel valleys of different heights and slopes which are separated by a saddle point. This is a case of bistability (for fixed intersystem distance). The problem is again analyzed first at the H F level, regarding the electronic relaxation vs electron delocalization conflict, and it is shown that for the heteropolar molecules one may have either a classical instability behavior (FH molecule) or a bistability behavior: in LiF the neutral UHF and ionic H F solutions are both stable between 6 and 46 bohr! The same "bistability" behavior may exist on the exact potential energy surface, as shown for the Liz + F problem. A final discussion points out the requirements to be satisfied in building bistable DA complexes, and suggestions are made accordingly.
-
-
I. Introduction When one excess electron or a hole migrates in a (supra) molecular architecture such as a molecular cluster An which is otherwise neutral, it may move between equivalent sites, through electron-transfer integrals which favor its delocalization. However one must remember that (i) this charge (electron or hole) creates an instantaneous field and this fluctuating field induces a dynamical polarization of the surrounding localized (or localizable) electrons. This is the electronic relaxation phenomenon.' (ii) When it is located on subsystem A,, the subsystem A: or A; would prefer to adopt another geometry than when it is neutral; the bonds usually are lengthened in the ions. This is a nuclear relaxation, which is well documented in the theory of electron transfer (see for instance refs 2-6). Within the HartreeFock (HF) approximation, one only takes into account the mean static field. If the hole is delocalized, the mean field that it creates is weak everywhere. While if it is localized, it creates a strong static field, and the electronic relaxation is taken into account in the mean field approximation. The HF description is the subject of a conflict between the delocalization of the hole and the electronic relaxation. For symmetrical geometries there are different regimes of the delocalization/relaxation ratio, which correspond to either delocalized symmetry-adapted solutions (when delocalization prevails) or localized symmetry-broken solutions (when electronic relaxation prevails). Regarding the nuclear relaxation, and considering only the electronic Hamiltonian, i.e., the Born-Oppenheimer approximation, things are quite similar. As a function of both the intersystem distance and the internal geometrical distortions of the subsystems, the potential energy surface (PES) may present either symmetrical minima (or valleys) corresponding to the situation Universite Paul Sabatier. *On leave from Laboratoire de Chimie Thtorique, Facult6 des Sciences, Rabat, Morocco. LOE-CEMES. "On leave from Laboratorio de Quimica Fisica, Universitat Autonoma de Barcelona, Barcelona, Spain.
in which the charge is delocalized or symmetry-broken minima in which the charge is essentially localized in one subsystem with adapted internal geometry. The two phenomena are very similar, ruled by isomorphic equations, so that the H F symmetry-breaking may be viewed as more than a purely academic problem, but as a relevant metaphor of a more physical phenomenon concerning the nuclear relaxation. This is fmt illustrated by the Az+problem (where A is a molecule). Section I1 is devoted to the molecular dimer cation problem, taking (H2)2+as a pedagogical example. The H F symmetry-breaking is studied in the rectangular geometry, while the nuclear relaxation phenomenon is studied as the rectangletrapezoid transition as a function of the intermolecular distance. It is shown that both symmetry-breakings are ruled by the same condition in terms of transfer-integral/relaxationenergy ratio, and that it behaves as a bifurcation, with a single critical distance where the symmetrical valley becomes a crest. For another electron-transfer problem, namely, the donoracceptor (DA) problem and its change from a neutral DA to an ionic D+A- situation, things are somewhat different, as shown in section 111. One may still think in terms of a conflict between the above-discussed forces, but there is no longer symmetry, and the phenomenon may be viewed as the matching of two valleys, which may be either continuous, through a more or less orthogonal or oblique zigzag, or discontinuous, with the coexistence of two (1) This is true in atomic clusters, such as those of He, Mg, or rare gases. For He,+, see: Rosi, M.; Bauschlicher, C. W. Chem. Phys. Le??.1989, 159, 479. Tarentelli, F.; Cederbaum, L. S.; Camps, P. J. Chem. Phys. 1989,91, 7039. Kabbaj, 0. K.; Lepetit, M.B.; Malrieu, J. P. Chem. Phys. Lett. 1990, 172, 483. For Mg,+ see: Durand, G.; Daudey, J. P.; Malrieu, J. P. Theor. Chim. Acta 1988,74,299. Durand, G . J. Chem. Phys. 1989,91,6225. Reuse, F.; Khanna, S.N.; De Coulon, V.;Buttet, J. Phys. Reu. B 1990, 41, 11743. (2) Dogonadze, R. R.; Kuznetsov, A. M.; Maragishvili,T. A. Electrochim. Acra 1980, 25, 1. (3) Sutin, N. Prog. Inorg. Chem. 1983, 30, 441. (4) Newton, M. D.; Sutin, N. Annu. Reu. Phys. Chem. 1984, 35, 437. (5) Mikkelsen, K. V.;Ratner, M. A. Chem. Reu. 1987, 87, 113. (6) Farazdel, A.; Dupuis, M.; Clementi, E.; Aviram, A. J. Am. Chem. Soc. 1990, 112,4206.
0022-365419212096-2162$03.00/0 0 1992 American Chemical Society
Instability vs Bistability in Electron Transfer
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2163
valleys in a whole domain of distances between D and A. The phenomenon is again illustrated first at the HF level on the LiF molecule for which stable solutions exist between 6 and 46 bohr, one being an ionic restricted HF (RHF) solution, while the others are two equivalent UHF solutions of neutral character. Regarding nuclear relaxation, the coexistence of two valleys is illustrated by the LizF problem in a triangular geometry. Section IV discusses the preceding results along two directions: (i) the HF symmetry-breaking, which is usually presented as an instability problem, but can also take the form of multistability; (ii) the possible occurrence of bistability in specifically designed molecular structures.
I
11. Electron Transfer in a Dimer Cation A. The Two-State Analytic Model. Consider a two-state
problem, involving two interacting configurations 4, and coupled with (&,)f&$b) = h. The Hamiltonian reduces to a 2 X 2 matrix f&,
I*
co.w14L>
Figure 1. Behavior of the symmetry-breakingparameter X, as a function of the hopping integral vs relaxation energy ratio.
= (4bl44b)
two opposite symmetry-broken solutions. Notice that eq 1 defines a half-circle, so that
and let one assume that E, and Eb Eb
are quadratic functions of an as yet unspecified parameter A, such that E,(X) = Eo
+ (A2 - 2 X ) u
&(A) = Eo + (A2
+ 2X)AE
For X = 0 the two configurations +,(A) and &(A) are degenerate, with the energy Eo.The minimum of &(A) is obtained for X = 1, and
= E,(1) = Eo - AE
X=1
- &q
= -1
similarly &bin)
= ~ ~ ( - 1= )
The lowest eigenvalue, solution of the secular equation
[
:+(A2-2k)AE-X
h E,
+ (A* + 2 A)AE -x
is
and if one neglects the A-dependence of the coupling element dh/dX = 0 the extremes of energy are given by
4XAE2
dX
which admits three solutions, namely, X = 0 in any case, for which, if h < 0, as usual for transfer integrals, = (& + 4b)/2’l2 (symmetrical solution), and eventually for
*
’,A
(4AE2- h2)/4AE2
lalhkl- 1 - O D
Ilh/ZAEpl-
alh/2Ml
1
=o
lh/ZAEI=O+
This is a typical bifurcation; the wave function is continuous around lh/2&qI = 1 but its derivative is discontinuous, so that the energy and its first derivative are continuous but the second derivative of the energy is discontinuous. In the above scheme, if one considers the surface of the energy as a function of two parameters X and h/2AE and if one starts from the strong coupling limit (JhI2i.U = m) and decreases h, one follows a symmetry-keeping valley up to the critical point, where two symmetry-breaking valleys start perpendicularly to the preceding one, leftward and rightward. Along the symmetrykeeping (A = 0) direction, the preceding valley becomes a crest. One might wonder whether another topological configuration would be possible, namely, the coexistence of three valleys in a certain domain of values of Ih/2AEl < 1. This is only conceivable if the Hamiltonian is made more complex. One might first drop the assumption that h is X - independent. For symmetry and continuity reasons (at least in the absence of an external field), (dh/dh)A,o = 0. Then if h = ho + aX2 ...,one might imagine that if a is positive (i.e., if the amplitude of the coupling increases with A), the symmetrical valley still exists for values of (h/2AE( < 1 , since the symmetrical solution takes the whole benefit of the resonance. However, this is not true, as is immediately seen from the degree of the eigenequation polynomial, which stands at degree four when h is only quadratic. The possibility of coexistence of two symmetry-broken valleys and one symmetry-adapted valley requires a sixth degree polynomial. We have explored another direction, which realizes that possibility, by considering “anharmonicity” terms in the X - dependence of the diagonal terms.
+
X = Eo + X z A E - d4X2AE2+ hZ
dx _ - 2XAE-
v
I
h
lhl. There are two symmetrical symmetry-broken solutions. The parameter A may be taken as a measure of the symmetry-breaking. It is easy to see that if that corcondition is satisfied the solutions A, = *(1 - h2/4AE2)1/2 respond to minima while the point X = 0 becomes a saddle point between these minima. Now one may see (cf. Figure 1) the behavior of the symmetry-breaking parameter A, as a function of lh/2AEI. For large Jh/2AEI,there is a unique symmetrical minimum. The relation Ihl = 2AE defines a critical point. When Ih/2AEl < 1, one finds
E(X) = Eo + (A2 - 2XAE)
+ a(X - 1)3
The appearance of a domain of Ih/2AEl (Le., a region of intersystem distances since (h/2hEl is a decreasing function of r) where three valleys coexist is actually possible when CY is positive and sufficiently large, but in that case the polynomial expansion becomes rather meaningless. One may say that for the two-state resonance problem the simultaneous existence for a given intermolecular distance of a symmetry-adapted and two symmetrybroken minima has no chance to occur. This does not exclude the possible existence of symmetry-broken minima and of a symmetrical minimum a t a different and significantly shorter intermolecular distance. Up to now, the nature of A has not been specified; two cases will be considered.
Durand et al.
2164 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
B. Electronic Relaxation: Hartree-Fock Symmetry-Breaking. 1. Theory. Let us consider the cation of a dimer A2+, where A is a closed-shellsystem (atom or molecule). For sake of simplicity we may restrict our development to the systems where A bears two electrons (He atom or H2 molecule), but the development would be valid as well if more electrons were involved. If A is a diatom (or more generally a molecule), we only consider geometries of the A2 problem where A and its partner (let us denote it B) are mirror images. Then the exact solutions of the electronic Hamiltonian H are symmetry-adapted. At the restricted Hartree-Fock level one may optimize a symmetry-adapted solution 4o = lggul where g and u are gerade and ungerade MOs, each of them having equal amplitudes on A and B. Then we may define
a = (g + u ) / f i
and b = (g - u ) / f i
which have large amplitudes on systems A and B, respectively. One may notice that if
4, = la661
f$b
= laiibl
these two determinants have one hole on A and B, respectively, and that 40
= (4, + 4 b ) / f i
i.e., the symmetry-adapted determinant, incorporates the resonance between two determinants bearing the hole on either the left or the right subsystem, but described with the unique set of MOs a and b. Notice that these MOs are neither optimal for 4, nor for 4b; they are a compromise. a is halfway between the best orbital for A in the presence of B+ and the best orbital for A+ in the presence of B. Now we may desire to optimize $, provided that excitations to ii, which is empty, are forbidden, in order to maintain an A+B situation. One may localize the virtual MOs a* and b*, maintaining the orthogonality constraints (a*la) = (a*lb) = (a*lb*) = 0 The MO optimization of 4, will proceed through MO mixings
E,(X=O)
- E,(X=l)
Le., the energy stabilization (positive quantity) that one obtains when relaxing the MOs of 4, subject to the constraint of a zero occupancy of ii. Then it is easy to see that if one uses the same MOs a’(A), b’(X) to describe f$’b(X) = la’(A)ii’(X)b’(X)l
= Eo + 2XAE + X2AE Of course, 4 b is destabilized when one builds it with the MOs which are optimized for 4,. One may notice that Eb(X=1) = Eb(X=O) 3AE
+
The optimum of Ebcorresponds to A = -1. Then the research of a symmetry-broken H F solution 4 = Iq1@,(021 is equivalent to the research of $(A) = cos (q)#,(X) + sin ($o)4’b(l), i.e., to the solution of the 2 X 2 Hamiltonian spanned by the +’,(A) and I$’b(h) V,(h) $’b@)
I
Eo - 2ME + h2AE
Eo + 2hM -t k2&?
1
Now one may neglect the X dependence of (f$’,(X)lflf#)’b(A)) (4’a(X)“’’b(A))
= (4alq4b)
the problem reduces to eq 1, and the same conclusions are valid; namely, the HF solution symmetry-breaking appears when the electronic relaxation energy AE of the optimized localized configuration d’, with respect to the energy of 4,, halfcomponent of the symmetry-adapted configuration 40, is larger than half of the electron-transfer integral ( 4,lfl&) (in absolute value). 2. Numerical Illustration. We illustrate this analysis for the H4+problem in a rectangular geometry. The small side of the rectangle is fixed at the value r = (re(H2) re(H2+))/2where re is the equilibrium distance of H2 and H2+, respectively, which are equal to the exact ones at the SCF level ( r = 1.69 bohr). The long side of the rectangle (H2-H2 distance) is labeled R . The basis set is a classical 2slp basis. As is clear from Figure 2, the symmetry-breaking takes place at R, = 4.1 bohr if the calculation is an open-shell restricted Hartree-Fock solution. The process of symmetry-breaking is a bifurcation, as may be seen from the fact that (i) the symmetry-broken (SB) potential curve is tangent to the symmetry-adapted (SA) potential curve at the critical point and (ii) the SB net charges tend to equate when R R,+.Notice that, as expected from the preceding analysis 6@B/6R is discontinuous at R,since (6qSB/6R)&+# 6qSA/6R= 0 (cf. Figure 2b). We have checked whether the relation 2AE = h is verified at the critical point. The relaxation energy was calculated as follows: the radial relaxation (AEh) of the electronic clouds to fit the H2 and the H2+ situations is distance-independent and is easily calculated at infinite distance as
+
where Fa is the Fock operator relative to the 4, determinant. Remembering Brillouin’s theorem for do(which defined a Fock operator Fo),it is easy to see that
( J , and Jb being the Coulomb operators associated with the MOs a and b);i.e., a will contract into a’and b will dilate into b.
Also
notice that in 4,, b’will be distorted toward A (polarization by the hole on A). Each single excitation brings an energy lowering, which to the second order may be written Now if we put a damping factor X on the orbital relaxation, writing
-
AEiso= E(H2) + E(H2+) - EH,+SA
it is easy to see that
The relaxation energy also involves a polarization energy of H2 by H2+,which is distance-dependent and has been calculated as the stabilization energy of H2in the presence of 0.5 point charges at the position of a phantom H2+.
with hE = -chEaa* a*
+ Zb’A E b b .
AE is the energy lowering of the optimized configuration +’,(A)
= la’(x)6’(x)b’(h)l
For X = 1, since X = 1 corresponds to the optimal MO relaxation
AE(R)pol = E(H2+
+point chargca)
- E(H2)
and
M(R)=
m i s o+ a p o l ( ~ )
The ‘hopping” integral h is calculated as
2h(R) = E ( u , ~ u , ) E(u,uU2)
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2165
Instability vs Bistability in Electron Transfer SA 51)
-1.72
E fu.a.1
intersystem distance R and the opposite distortion 6r (and -6r) of the two bond lengths. Forty points have been calculated for 3.6 < R < 6.0 bohr and l&l< 0.27 bohr. The symmetry-breaking takes place for R = 4.3 bohr, and the phenomenon appears as a typical bifurcation (see Figure 3). The theoretical determination of R, by the study of the distance R where 2AE = h is rather simple. AE may be considered as distance-independent
AE = E(H2)r,(H2) + E(H2+)re(H3- E(H2)r - E(H2’)r
-1.74
and is equal to -0.0155 hartree. The “hopping” integral is easily evaluated from the energy separation between the 2B, ground state and the lowest excited state of symmetry *Alg 4.00
2h = E(’B,) - E(’AIg)
5 00
R Iu.a.1
0.00
SA 0.70
Figure 3c gives its evolution around the critical point and one sees that the theoretical relation, which in principle is only true for harmonic oscillators, is perfectly satisfied; h - 2AE for R = 4.28 bohr, while the potential surface presents a bifurcation close to R = 4.3 bohr.
-
111. Electron Transfer in Donor-Acceptor Complexes: The
CHARGE 0.60
sa
0.50
4.00
5 00
R lu.a.1
0.03
(u.a.1
D-A D+A- Transition A. Analytic Model for the Matching of Two Valleys. Let us consider a donoracceptor complex. Two states are in competition. The neutral state DA, which is only stabilized by long-range dispersion forces, and the charge-transfer state D’A-, which starts from the asymptotic higher energy (PID - EA, above the neutral dissociation) but is stabilized by R’Coulomb attraction. The distance R between D and A is a first geometrical parameter, but the optimal geometries of D and D+ (respectively, A and A-) are different and one may define a second geometrical parameter A, which varies between -1 and +1 and which represents the collective geometry change from D to D+ and A to A-. Whatever the internal coordinate i D of D or j A of A, we consider that
dE h/Z
0.01
j(X) = 4.00
5
R 1u.a.) Figure 2. RHF symmetry-breaking in the rectangular H4+problem as a function of the intermolecular distance: (a) energy of the symmetryadapted (SA) and symmetry-broken (SB) solutions; (b) atomic electronic population (for the SB solution, this population corresponds to the molecule that tends to become ionic and tends to I/*); (c) R-dependence of the relaxation energy AE and hopping integral h.
Figure 2c shows that the critical point takes place at the distance where 2AE(R) = h(R). C. Nuclear Relaxation. Section A described the phenomenon in the harmonic approximation, and we shall only perform here a numerical illustration. The simplest illustration of the symmetry-breaking phenomenon with respect to the nuclear geometry is again qbtained by treating the H4+problem and the rectangle trapezoid geometry change when the distance between the two centers of the H2 subsystems is increased. H2+is much longer than H Z : rc(H2+)= 1.97 bohr while re(H2) = 1.43 bohr, with r as the mean distance. It is clear that, at least within the Bom-oppenheimer approximation, at long intersystem distances the charge will tend to localize on one subsystem; and the corresponding interatomic distance will be long, the other one being short, H4+ = H2+-H2. At a short enough interatomic distance the geometry will be rectangular with a sharing of the positive charge between the two subsystems. The calculation has been done using full CI with a 2slp basis set, and the potential energy surface is drawn as a function of two parameters, the
-
jA+jAL
+ -A
( jA - - j A) L
Then the problem of the electronic ground state of the [DA] complex may be seen as resulting from the electronic interaction between two diabatic states DA and D+A-, the potential surfaces of which are two grossly parallel valleys directed along R, the first one (centered on A = -1) being rather flat (associated with DA), while the other one (centered on A = 1) is going down toward small values of R. The question is the following: under which conditions does the adiabatic potential surface, resulting from the coupling of the two diabatic states, present fwo different valleys for a certain domain of values of R , and what is that domain? To solve that problem,’-I0 one may assume that the two valleys have the same A-curvature (this is a rather crude approximation). Let us call AE the relaxation energy of DA (respectively D+A-) from A = 0 (i-e., for intermediate geometries of the partners, exactly halfway between those of DA and D+A-) to X = -1 (respectively + l ) . Then for a given value of R, one may write the 2 X 2 Hamiltonian wD04);t
IbO+ -:)4 I
[ y+2h)AE
c + (h2- 2h)A.E
1
where c is the energy difference between the bottoms of the two (7) Wong, K. Y . ;Shatz, P. N . Prog. Inorg. Chem. 1991, 28, 369. ( 8 ) Robin, M. B.; Day, P. Adu. Inorg. Chem. Radiochem. 1967,10,284. ( 9 ) Hush, N. S. Prog. Inorg. Chem. 1967, 8, 391. (10) Khan, 0.;Launay, J. P. Chemrronics 1988, 3, 140.
2166 The Journal of Physical Chemistry, Vol. 96, No. 5, 1992
E t 1 (u.a.1
Durand et al. EwlO ( u . a . 1
(a)
-0.m
(C)
-0.1
-0.29
-0.30
6.71
R.6.0 R4.6 R4.2
-0.31
-0.32
I R-4.8
-0.33
w4.4 -0.775 ,
-0.5
R-4.0
1 4.1
4.2
4.3
4.4
4.5
R tu.a.1
R4.6 4.7M
.
-0.795
.
1
I
I
1
0.054
1
0.270
0 . m
(u.a.1
br
E + I (u.a.1 -0*710
-0. BiO
(bl
1
i 1
I
-0.81
I
-O.u
I
I
0.00
-0.27
6r
1
0.27
1
0.u
1
0.a
(u.a.1
Figure 3. Symmetry-breaking under nuclear relaxation in the [H2-H2]* problem as function of the intermolecular distance R and the rectangle to trapezoid distortion from a mean H-H distance of 1.7 bohr (distances in bohr, energies in atomic units): (a) sections of the potential energy surface (PES); (b) overall view of the PES; (c) determination of the critical distance from the h = 2AE relation.
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2167
Instability vs Bistability in Electron Transfer diabatic valleys at that value of R. The secular equation gives the eigenvalue
E (u.a. 0.05
-
-0.00
-
y = - - X - b + x 2- l / 2AE
-0.01
-
u = h/2AE
-0.10
-
b = c/4M
-0.11
I
1 x = -[c
+ 2A2AE - l / ( c - 4AAE)' + 4h2]
2 Or taking 2AE as the unit of energy
15.0
with
Rc
and One may study the existence of one or two minima by studying the first and the second derivatives of y as a function of A. In order to have two minima, the second derivative should have two inflection points such that the first derivatives in these points have opposite signs. Since d2Y _ -1dA2
-3.0
-2.0
-1.0
6r
0.0
1.0
2.0
3.0
lu.a.1 (a)
(12
[(b - A)2
E (u.a.1
+ u2]d-
0.05
-
-0.00
-
-0.01
-
d2y/dA2 = 0 implies
[(b - A)2
+ u2]{-
= u2
(b - A)2 = u4/3- uz
which must be positive. This condition requires u2 < 1, Le., Ih/2AEI < 1. The two valleys can only exist if this inequality is satisfied; this condition is exactly the same as the one obtained in the preceding section. This is obvious since if two valleys exist, they exist for c = 0, which is the previously examined degenerate situation. The second condition can be expressed as
IcI < 2 [ ( 2 ~ U ) -~ /1h12/3]3/2 ~
lS.0
Rc
-0.10
-
-0.15
I
(2)
-3.0
-2.0
Then of course Icl < 4AE. The domain of coexistence of two valleys around the point R, (where the two bottoms are degenerate) is smaller that the domain defined by the condition Icl < 4(AEI, which by itself is rather limited since IAEI is usually small and the energy difference Icl between the two diabatic configurations depends strongly on R. One must remember that in the harmonic approximation 4 p E = Ip(vert) - Ip(adiab) - EA(VCrt) + EA(adiab) = A(1P) - A(EA) Figure 4 illustrates the three typical situations for the adiabatic matching between the R-' descending ionic valley and the flat neutral valley. B. The HartreeFock Discontinuity in the Neutral/Ionic Curve Crossing: Instability and Bistabity. Let us consider a strongly ionic molecule such as H F or LiF. At short interatomic distances, the best HartreeFock wave function is of closed-shell character, of the type H + F or L i + F , while at long interatomic distances the lowest solution corresponds to an open shell since one dissociates into H' (or Li') F'. In a minimal basis set the symmetry-breaking will necessarily take the form of a bifurcation; i.e., the restricted Hartree-Fock solution becomes unstable for R > &. In such a case there is no relaxation, the neutral and ionic configurations are represented with the same u and 17 orbitals so that the problem is essentially the same as for the H2 problem, which is treated in text books, except for the fact that the u MOs do not have equal coefficients on s (the 1s A 0 on H or 2s A 0 of Li) and on z (the pl-type hybrid of the halogen atom). The restricted H F solution will become unstable for R > R,, and at R, the symmetry-breaking behaves as a bifurcation. For n o n " a 1 basis sets, two different behaviors are possible. Figure 5 gives the evolution of the R H F and UHF energies for FH calculated in a conventional DZ+P basis set. The UHF solution appears for R > 2.6 bohr, and at R, the U H F solution 0 when R coincides with the R H F one since (4uHFlS21~UHf)
+
-
-1.0
6r
0.0
1.0
2.0
3.0
(u.a.1 (b)
-0.15
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
sr (u.a.1 (CI Figure 4. Model potential energy surfaces exemplifying the matching between an ionic D+A- and a neutral D A surface, for different ratios h / 2 A E of the charge-transfer vs relaxation energy, keeping a constant value for h and a R'variation of AE, The bottoms of the ionic diabatic valley are taken degenerate for R = 10 au: (a) h / 2 A E = 1.5, oblique matching into a single valley; (b) h/ZAE = 1, critical regime (single valley with a rectangular zigzag); (c) h / 2 A E = 0.5, coexistence of two valleys in the region of diabatic crossing.
-
R,+, as appears in Figure Sb. This is a standard bifurcation behavior. For LiF, the situation is quite different. A recent paper" has
2168 The Journal of Physical Chemistry, Vol. 96, No. 5, 19F'2
Durand et al.
E+24 (u.8.)
-O.=
-0.43
i P
I/
2.00
3.00
4.00
R (u.a.1
(b) Figure 6. Effect of electronic energy relaxation near the ionic/covalent avoided crossing of an ionic molecule. Thin lines represent diabatic states and thick ones represent adiabatic states starting from compromise (-) or from LiCF and Li'F' (- - -) molecular orbitals. (-e)
I
I
l
2.00
l
l
l
3.00
l
4.00
R (U.8.) Figure 5. HF symmetry-breaking for the FH molecule: (a) energies of the RHF and UHF solutions; (b) mean values of the S2 operator.
shown that CASSCF solutions (Le., multiconfigurational S C F solutions) were discontinuous in the region of curve crossing between the two basic configurations of the problem, namely, L i + F and LFF'. Let us recall that this curve crossing is expected at a distance close to
R;' = IP(Li) - EA(F) since the neutral (Li'F') diabatic potential curve is supposed to be very flat as long as overlap is negligible, while the diabatic ionic potential curve (Li+F) starts from an asymptote situated at IP(Li) - EA(F) and is attractive, due to the R'Coulomb interaction. If the exact ionization potential and electron affinity are used, the curve crossing is expected to occur at R, = 12 bohr, a distance where the matrix element between the two configurations (4LiFlMdLi+F)
= (2sLilF12pzF)
is really small (( 0.1 eV). At the SCF level the ionization potential of Li is almost exact (except for negligible core valence correlation effects), while the EA of F will be significantly underestimated even if large basis sets are used, since the correlation energy is larger in F than in F. The asymptotic estimate of the HF electron affinity of F is 2.65 eV. The curve crossing at the S C F level is then expected near 1.5 bohr, and at this distance the hopping integral ( 2sLilF12pz,)remains small. Now one must remember that the S C F AOs of F are significantly more diffuse than those of the radical F'. One should also (11) Bauschlicher, C . W.; Taylor, P. J . Chem. Phys. 1988, 89, 4246.
notice that at finite distance in the ionic configuration the AOs on F will be polarized by the field created by Li'; this polarization mixes for instance the 2p, A 0 with d AOs. This is an electronic relaxation. If one started from compromise MOs, halfway between the best MOs for Li'F' and for L i + F the electronic relaxation energy for each form, Le., the energy stabilization obtained in optimizing LI'F' or L i + F by M O relaxations of opposite directions, would be large. If one calls j , the compromise MOs (those given by a stateaveraged MCSCF calculation on the two lowest states) and (PN and (m the optimized MOs of the neutral and ionic configurations (inhibiting the charge transfer), the electronic relaxation energy is = EN(*) E,(?) - Edcpl)
with EN(+) = (~N(+)IHI~N(+)) etc.
and AE is expected to be much larger than h. Hence at R = R,, as clearly pictured in Figure 6, one gains much more energy by sacrificing the charge-transfer interactions (treated to the second order as h2/4AE)and taking benefit of the full relaxation energy than by freezing the MOs at their compromise picture, which ensures degeneracy, and allowing the resonance between the two configurations. The problem has been treated analytically elsewhere in the two-configuration SCF formalism.12 The RHF/UHF dilemma is so close to that case that it does not deserve to receive an almost identical algebraic derivation. It is easy to show that if at the curve crossing point R, where
( ~ N ( ? ) I M ~ N) (=~(d4j,)IWdj,) ,) 12pEl
)
> 1(4N(?)Iflh(?))l
(i) the MCSCF solution is discontinuous, jumping from an essentially ionic solution for R < R, to an essentially neutral solution for R > R,. (ii) There are two stable solutions, an U H F one, of (12) Sanchez de Meras, A.; Lepetit, M. B.; Malrieu, J. P. Chem. Phys. Len. 1990, 172, 163.
The Journal of Physical Chemistry, Vol. 96, No. 5, 1992 2169
Instability vs Bistability in Electron Transfer
1.25 atomic units, 4 times smaller than the R H F dipole moment at the same distance (5.7 au). One may easily have a direct estimate of the distance R” at which the RHF solution becomes unstable. When lhl h was proposed by Shaik and W h a n g b ~ . ~ ~ Registry No. H4+, 12184-94-0; LiF, 7789-24-4; Liz, 14452-59-6;F, 14762-94-8; HF, 1664-39-3. (21) Boschi, R.; Clar, E.; Schmidt, W. J. Chem. Phys. 1974, 60, 4406. (22) Treboux, G., private communication. (23) Shaik, S.;Whangbo, M. Inorg. Chem. 1986, 25, 9201.
Kinetics and Mechanisms of the Reactions of CH,S, CH,SO, and CH,SS with 0, at 300 K and Low Pressures Florent Dominc, A. R. Ravishankara, and Carleton J. Howard* National Oceanic and Atmospheric Administration, ERL, R/E/AL2, 325 Broadway, Boulder, Colorado 80303, and the Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado (Received: July 19, 1991; In Final Form: October 30, 1991)
-
-
-
The reactions of CH3S + O3 products (l), CH3S0 + O3 products (2), and CH3SS + O3 products (3) were investigated at 300 K in a discharge flow tube reactor coupled to a photoionization mass spectrometer. The measured value of kl is (5.7 f 1.4) X cm3 molecule-' s-' in 1 Torr He. We observed that OH was produced in this reaction or in subsequent steps and that complex branched chain reactions, which generate CH3S from its precursor molecule, took place in our flow tube reactor. We found that CH3S0 was a product of reaction 1 and the branching ratio for this channel was 15 4%, between 0.7 and 2.2Torr He, independent of pressure. A preliminary value of k2 = (6 f 3) X lo-" cm3 molecule-l s-I was measured. CHJS was not a major product of reaction 2. These results suggest that the reaction with O3is a major CH3S removal process in the atmosphere. The rate coefficient for the reaction of CH3SS with O3(3) was measured to be k3 = (4.6 1.1) X lo-" cm3 molecule-I s-l.
*
*
Introduction
Dimethyl sulfide (CH3SCH3,DMS) is estimated to make up between 50 and 90% of the total biogenic sulfur released to the atmosphere.' The CH,S radical is thought to be an important intermediate in the atmospheric oxidation of DMS and of other biogenic sulfur compounds, such as methanethiol (CH,SH) and dimethyl disulfide (CH3SSCH3,DMDS).z4 Therefore, several groups have investigated the reactions of CH3S with the abundant Three groups have tropospheric oxidants 02,NOz, and 03.5-9 reported rate coefficients for the reaction of CH,S with NOz+' and obtained results in the range (5-1 1) X lo-" cm3 molecule-' PI. Even though this reaction is fast, there generally is not enough NOz in the marine troposphere, where most of the DMS, and hence CH3S, is present, to make this reaction an important CH3S *Author to whom correspondence should be addressed at NOAA.
removal. An earlier measurement suggested that the reaction of O3 with CH3S was slow* 58 X cm3 molecule-I s-I. The reaction of CH3S with O2is very slow and an upper limit has been (1) Andreae, M. 0. In The Biogeochemical Cycling of Sulfur and Nitrogen in the Remote Atmosphere; Galloway, J. N., et al. Eds.; D. Reidel: Dordrecht, 1985; pp 5-25. (2) Niki, H.; Maker, P. D.; Savage, C. M.; Breitenbach, L. P. Int. J. Chem. Kinei. 1983, 15, 647. (3) Hatakeyama, S.; Akimoto, H. J . Phys. Chem. 1983, 87, 2387. (4) Grosjean, D. Enuiron. Sci. Technol. 1984, 18, 460. ( 5 ) Balla, R.J.; Nelson, H. H.; McDonald, J. R. Chem. Phys. 1986, 109, 4 - 3
IUI.
(6) Tyndall, G. S.; Ravishankara, A. R. J . Phys. Chem. 1989, 93, 2426. (7) Domini, F.; Murrells, T. P.; Howard, C. J. J . Phys. Chem. 1990.94, 5839. ( 8 ) Black, G.; Jusinski, L. E. J . Chem. Soc., Faraday Trans. 2 1986.82, 2143. (9) Tyndall, G. S.; Ravishankara, A. R. J . Phys. Chem. 1989, 93,4707.
0022-365419212096-2171 $03.00/0 0 1992 American Chemical Society
