The Journal of
Physical Chemistry VOLUME 98, NUMBER 47, NOVEMBER 24,1994
0 Copyright 1994 by the American Chemical Society
LETTERS Equilibrium and Nonequilibrium Solvation and Solute Electronic Structure. 4. Quantum Theory in a Multidiabatic State Formulation Roberto Bianco, Jesus Juanos i Timoneda, and James T. Hynes* Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-0215 Received: August 8, 1994; In Final Form: October 11, 1994@
A theory of the solute electronic structure for chemical reaction systems in solution [Kim H. J., Hynes, J. T. J . Chem. Phys. 1992, 96, 50881, in terms of a solute description via two chemically relevant valence bond (VB) states and a dielectric continuum solvent model, is generalized to account for more than two solute VB states. This extension is required for a number of important solution reaction systems. The electronic polarization of the solvent is quantized via a coherent states treatment, and its orientational polarization spans both nonequilibrium and equilibrium solvation regimes. The nonlinear Schrodinger equation and system free energy are obtained in a solvent coordinates framework.
Introduction Recently, Kim and Hynes' have presented a theory for the electronic structure of a solute system in solution from a twosolute diabatic VB state perspective. This theory, which treats the solvent as a dielectric continuum, allows for nonequilibrium solvation of the solute by the slow solvent orientational polarization-a feature necessary for the description of chemical reactions-and includes a quantized description of the highfrequency solvent electronic polarization over the entire range of electronic resonance coupling between two solute VB states. The solute boundary conditions are treated in the dielectric image approximation.2 As noted above, the theory was limited to a two VB state description, adequate for a wide variety of solution chemical reaction issue^.^ However, several important chemical systems evidently require more than two VB states for their description. For example, the [CH3C12]- Sp42 reaction ~ y s t e m can ~ ~ .be ~ viewed as comprising the three VB structures CI-/CH3-C1, C1CHdCl-, and Cl-/CH3+/Cl-, where the slash means "no bond". The Menschutkin reactions H3N CH3X H3N-CH3+ X(X = F, C1, Br, I) require at least three related str~ctures.~ A similar framework applies to acid-base proton transfers, which
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@Abstractpublished in Advance ACS Abstracts, November 15, 1994.
0022-3654/94/2098- 12103$04.50/0
may be comprehended in terms of the three resonance ~ t r u c t u r e s A-WB, ~ ~ ~ ~ . ~A-/H+/B, and A-/H-B+. Three (or more) VB states are also invokeds to explain the commonly observed breakdown of the Bronsted rate-equilibrium constant relationship. More generally, more than three important VB states are thought necessary to describe the electronic structure of such important reaction species as H30+, NH3, and 0 3 . 9 Here, we obtain the general form of the n-state free energy and derive the nonlinear Schrodinger secular equations. In addition to allowing the burgeoning vacuum diabatic VB electronic structure development^^^^^^^^ to be exploited for solution reactions of the types above, the present treatment provides the starting point for a theory in which the solute cavity-solution boundary conditions are more rigorously accounted for."
Free Energy with Quantized Electronic Polarization The system Hamiltonian operator, with quantized electronic polarization, is1
H = Ho + l/zM,,-l[&e,lI&el] + 1/2Cel[PeI IP,] + 4~t[+,tJ P O ~ I - @e + P O ~ I I E O I + '/2Cor[Por1 I P ~(1) ~I in the dielectric image approximation, with &J the vacuum Hamiltonian operator, Mel the mass associated with the electronic 0 1994 American Chemical Society
Letters
12104 J. Phys. Chem., Vol. 98, No. 47, 1994 polarization, Re]the momentum fieldAoperatorconjugate to the electronic polarization operator Pel, Eo the solute electric field operator, and Po, the orientational polarization. We have defined the constants C1, = E . & ~ and Cor= (1 4nxOr)/xor, with the electronic and orientational susceptibilities given by E.. = 1 4 ~ x eand E = E4zxor,in terms of the solvent static and high-frequency dielectric constants, E and E,. We have also defined the integrals [AJIB] Jvdr A(r).B(r) over the solvent volume V. The coherent state wave function is of multiconfigurational form'
+
+
+
sz (iJEob)Afori f j , with (6), since the off-diagonal exchange fields (ilE0b) can be usually considered smal1.l~~G is a minimum when 6G is equal to zero, which requires (j f i indicates a single sum)
This corresponds to the relation between the electronic polarization and field components
NS
10)= ~cili)lP:)); i=l
N,
with the inverse of an N, x N, polarizability matrix {a,} given by
+
a,.-' rl = 1 / 2 ~ e , [ d , ( ~ ~c;i C c g i k )- (1 - ~ , ) C , C , Q ~ I (11) where the {li)} are solute diabatic VB states and the {IP!')} are solvent electronic polarization coherent states. The orthogonality is guaranteed by the orthonormal {li)] set.12 Here and henceforth all sums run from 1 to N,. The coherent states satisfy'
p,Ip:)) = p!)lp:))
In particular, the P!) are the eigenvalues of the electronic polarization operator such that the c value electronic polarization is P, = Zic?~!). With (1)-(3), the free energy is obtained as
Hi
with 60 the Kronecker delta and the Q,,
e, defined by
2Hf, hoe,
--
(H'y is typically negative.') The {e#) compare the relative time scales of the solute and solvent electrons' 2(Hfij(gauges the interconversion frequency between the solute VB states, while we] gauges the frequency of the solvent electronic polarization and is approximately related to the solvent electronic absorption spectral peak.14 Before substituting (10) into G (4), we regroup the terms nonlinear in {P!'} in (4) [cf. eq 91 to obtain
G = (@\kl@)
+ ctcj(Py)lP!))Hf, + I/2celcc;[P:)I (P:'] +
= C C ; { H ; - [P~)l~(ilEoli)] - [Porll(i\Eoli)]} I
i
ij+i
4~~ci2[PO IP2)l r l + 1~2cor[po,l IPo,1 (4) i
,:
+ Pz'l I(ilEob)]- [(ilEob)l IPor] (5)
with Hi = (i(&io(j) the vacuum contribution. To find the functions {P!'] which minimize G, i.e., equilibration of the fast electronic polarization, we first approximate'
(p!'Ip,o")
1 - Y[p"' - 0') (i) e P, IIP, - P,o"l;
Y = Ce1/4hwe1(6) The second term in (6) is 6G = G[ ..., Pp' QP!',
+
which when substituted together with (8) for the {P!'] in (4) yields the system free energy with equilibrated quantized electronic polarization as
G =(
where the off-diagonal solvent-renormalized electronic resonance coupling elements are
H', = H - 1/2[P!)
i
Then the functional variation
-
~ I ~ ~ O I W [WIEOIWI
1por1
+ '/2~01~0r11por1 + Ge (14)
where l(Y) is here and hereafter the effective solute wave function i
i
with equilibrated solvent electronic polarization, (YI&ojOlY) is the vacuum-like energy contribution, the second term is the solute interaction free energy with the solvent orientational polarization field POI,and the final volume integral is the solvent orientational polarization self-free energy term. The electronic polarization contribution to the total free energy is
...I - G[ ..., Pf), ...I reduces to
QG = [6Pf'I IC,,C,'P:' - ~ Y C ~ ~ C F ' &-PP!':) ) - gi] (7) whose explication will now concern us. The self-consistent ( S C ) limit1315involves a time scale of the with the components Ci of the general field C = Xi&i defined solvent electrons slow compared to those of the active solute by electrons: {eij}>> 1. In this case, aij eel-'l6 [cf. (10) and (1 l)], and each of the r-dependent components Pf) is identical gi= ci(il81Y); 6 E E, - 4nPOr (8) to a clas$cal electronic polarization field P?) = P,o"= P, = To obtain (7), we have further approximated' (ilE~b)(P~)lP~') Cel-'(YICIY) = C,l-'&, and G, becomes pi
A
A
-
Letters Gsc = - 1 / 2 C , l - 1 [ ( Y ~ ~ ~ Y ) ~ ~ ( Y ~(17) ~~Y)] which involves the square of the average field. In the opposite, Born-Oppenheimer (BO) limit,1317-19the solvent electrons are fast compared to those of the solute, in the sense that {ey} 0; then (11) reduces to aU-l= C,JC&, such that each electronic polarization component satisfies cfPf) = Cel-lgiir and Ge becomes
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G,BO
= - 1 / 2 ~ e 1 - L J d r(YIPIY)
(18)
which involves instead the average of the square field. Note that the full limiting BO and SC free energies differ only in their electronic polarization contributions. Our next step is to recast (14) for the free energy as G = GBo
+ G, - G':
(19)
in preparation for exposing its limiting SC and BO forms. Here the total BO contribution, from (18) and (14), is' GBO= ( W f i O l W - [ ( ~ l E o l ~ ~ I I P o , 1 + 1/2Cor[PorlIPorl
+ G;O
(20)
With the resolution of the identity Cili)(il = 1 on the {li)} subspace, this can be written as a sum of partial contributions from the states:
ij
+
GF(ij) = - ' / 2 C , , - ' { ~ ~ [ ( Y ~ & ' ~ ) ~ ~ ( j ~ ~ ~ Y ) ]
C,2[(W&)l I(il a w l 1
where we also have the sum property for the fluctuation term
(21)
To properly combine G, and G F in (19), it proves important to make use of the property of the polarizability matrix
The explicit definition of Gsc follows from (27) and (20) for eo.Note that the Po,contributions in G," cancel and that the diagonal component Gt(ii) = 0. We now rewrite the free energy (26) one final time by making use of the general form of the polarizability matrix aij = (- ly+j(Mji/D),where Mji and D are a minor and the determinant of {aij-I}, respectively. The minors {Mji} and D all have in common a term, here called F [cf. Appendix], which does not depend on the indices. Thus, the aijhave a term in common. This critical characteristic leads, through (26), to our final free energy equation G = GBo
+ 1/2Cel
(aij - a)Gt(ij)
+ A(GSC- GBo) (28)
ij t i
where A = 2(ce1/2)Ns[n~,ck]F/D, and = ~ ~ 1 is- the l ~common term for the {aij}. The central sum in (28) vanishes in both the SC limit (where aij+. a) and the BO limit (where both aij, B 0), while the limiting 2 values are ,IBo= 0 and Asc = 1 [cf. Appendix]. Thus, (28) gives the proper limiting free energy functions. In the special case of two states, the sole off-diagonal element simplifies to aI2= B = Ce1-'[el2/ ( 2 ~ 1 ~ 2~12)], the central term vanishes, and the simple additive result G = A(GSCof ref 1 is recovered.
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+
eo+
eo)
Nonlinear Schrodinger Secular Equations which can be verified from (1 1) after inversion, which leads to G':
= -1/2zc:aij[(Y@li)/
I(il$lY)] = ' / 2 C , l ~ a i j G ~ o ( i j )
ij
ij
(23)
With (8), (16) for G, can be written as a sum involving partial SC free energy terms
G, = 1 / 2 C , , ~ a , G ~ c ( i j ) ; ij
GF(ij) = - C , l - l ~ i ~ j[(Yl&'li)ll(jl&lY)]
(24)
The expression of (17) for G F in a form consonant with G': (21) is easily obtained as G F = '/2cG3ij)
(25)
ij
Now we can express the free energy in terms of its limiting SC and BO forms, namely
The solute electronic structure in solution is determined by solving the nonlinear Schrodinger equation obtained by the minimization of the free energy (28) with respect to the state coefficients (c;} in (15), subject to the normalization constraint, (a/ack)[G - E(&? - l)] = 0, which yields the secular equations C/HkrCr = ECk ( k = 1, ...,NS),where E is an effective energy eigenvalue,'," and with the symmetry property Hkl = Hik. To obtain the Hkl, we require that each free energy component derivative be expressed as the sum aOlack = ClClOkl, with 0 k l = Olk. From (28), we obtain aG - aGBo
ac,
ack
+
e
1/2~el ijti
+
(aij - a)ack
The BO contribution is aQo/ack = 2ClH:
CI,
with
H ! : = (k1fi010- 1 / 2 ~ e ~ 1 ~( ~ v dI E r ~ -I'[(~IEOIOI o
1~011
6-
(30) G = GBo
+ '/,C,,z
a,Gt(ij) ij t i
(The special BO limit result is closely related to that of Basilevsky et al.,'* although the expansion basis differs.)
Letters
12106 J. Phys. Chem., Vol. 98, No. 47, 1994 The derivative of the fluctuation term is aG;(ij)/ack = 2E1G;'~'(ij)c/,with A kl Ge '
ce1-l (u)= 2 [ciGIEoly)
dikGlkoll) - d j k ( i l E o l l )
6ijkl
HfY = (klfioJZ)- 1/2Ce,-1Jvdr (kJE;IZ) - s(kl)
- c j ( i ~ E o l y ) ~ 16ijk/l
+ d&lE,lk)
Q0 are expanded. The use of (8) for 2 leads to the final working equations for those contributions:
(36)
+
H', = H:, - 1/2Cc,[ai1 a j ~ ~ [ ( l l E o l(ilEOb)l l~)l -~ ( i j ) - djl(ilkolk)
I
(37)
(31) The derivative of ai, is expressed as the sum
The Po, contribution in G in (28) appears solely in (20) can be written as
Q0,
and
+
GBo = (Y lfi,,lY) - Ccicjs(ij) ij
with the matrix elements a: for three states reported in the Appendix for illustration. With these results, we can now express the Hamiltonian matrix elements in their final form
m=l n = l
-.
C [(ao - a)GfSk'(ij)+ ayGf(ij)]
H i = '/zCel
(33)
ijti
- HE:" = (1/2)Ci,Jti with HfY defined in (30) and GY'(ij). As with the free energy, the Hamiltonian matrix elements are expressed in terms of limiting SC and BO forms and correctly reduce in the two limits. In the nonequilibrium regime, the solvent orientational polarization is arbitrary, and the infinite manifold of its degrees of freedom is conveniently compacted into a, set of solvent c o o r d i n a t e ~ ' . ~defined ~ J ~ . ~ as ~ s ( i j ) (l/c,)[(ilEOb)llPo,] (i, j = 1, ...,N J , where each field component (ilE0b) corresponds to a single solvent coordinate s(ij). This definition is connected to G and Hkl via the ansatz that the nonequilibrium orientational polarization is a linear combination of equilibrium components,',' 1,18each equilibrated to one of the bare solute electric field components (iJEob): Po, = Cf;f=, P,PE;~", where M = N,(N, 1)/2; the nonequilibrium character is exclusively attached to the arbitrary weights bm}. The index m can be better understood by considering the equilibrium form of Po,,
+
M
PZ = Cw",":y
(34)
where G :: is defined by (21) via the substitution 2 EO. In the nonequilibrium solvation regime, the nonlinear Schrodinger equation is solved by an iterative, self-consistent algorithm, by chosing arbitrarily the solvent coordinate s(ij) values. After solving the Schrodinger equation, the {c,}dependent terms in the free energy (28) can be computed. In the equilibrium regime, the orientational polarization is proportional to the solute bare electric field [cf. (34)], and the Pardependent terms are no longer arbitrary; they have to be recomputed at each step in the iterative cycle. An implementation of these algorithms will be illustrated in ref 20.
Acknowledgment. This work was supported in part by NSF Grants CHE88-07852 and CHE93-12267. R.B. gratefully acknowledges financial support through the Dottorato di Ricerca program of the Universitfi di Pisa, Italia. Appendix The general formula via (11) for aij in (10) is not available in compact form. We provide the analytical formula for three states and discuss critical points in the strategy to obtain its derivatives. The description below is sufficient to characterize the cases of more than three states and reduces, in the two state case, to the results of ref 1. We only need the off-diagonal elements of aij, since G;(ii) = 0 [cf. (26)]. The off-diagonal cofactors Cji = (-I)i+jMj; of {au-'}in aij= Cj;/D for three states are Cji = Q[2cgij
m= 1
which is equal to ~oJ~)(Y!lE~lY), with the equilibrium weights mapped according to1
'
w: = (2 - dkl)ciqc7q; m =~ , ( k 1) - l/,k(k - 1)
Q E (C&)
2
c1c2c3;
F
ijk = 123, 132,231
E C1@12@13f '2@12@23
+
c3e13e23 (A.1) with the determinant of
+I
+ FJ;
{ a i j - l }equal
to
(35)
The nonequilibrium analogue Po, is simply obtained by replacing the {w:} with the bm}. The solvent coordinates can then be M written as the scalar product'' sn = Em=lcdm, where = (l/c-)[(ilE~~)l JP:.'"] is a symmetric, invertible matrix, and n is mapped from (i, j ) via the analogue of (35). The solvent coordinates enter the Schrodinger equation both through the (30)-most importantly-and the off-diagonal aij [cf. Appendix], which contain the solvent-renormalized electronic coupling H'u (5). They also appear in the free energy when the integrals of the orientational polarization in (20) for
cm
The common term F is obtained, for any number of states, by dividing by (Cel/2)Ns-1nr&ckany diagonal minor M',* of the matrix which results from erasing the 2c12terms from the diagonal elements of {a,-'} (11). For large {eu},only the common a, factor survives, such that Asc = 1, as noted below (28). For the a, derivatives, it is important to cast the {eu}factors in terms of the fractions f, erl/(2cIc, el,), such that the required off-diagonal avcan be written as
+
Letters
with the index triplets ijk for N,, given in (A.1). In general, the derivative of aijis
Here we focus only on the nonstandard portion of this, which involves the fil derivatives
With the solvent-renormalized coupling (37), by expanding the wave function IW) and then using (22) to cancel the terms involving Por,we obtain
J. Phys. Chem., Vol. 98, No. 47, 1994 12107 (4) (a) Shaik, S. S.; Schlegel, H. B.; Wolfe, S. Theoretical Aspects of Physical Organic Chemistry: f i e sN2 Mechanism; Wdey: New York, 1992. Pross, A.; Shaik, S.S.Acc. Chem. Res. 1983,16,363. Shaik, S. S. J . Mol. Liq. 1994,61,49. (b) Sini, G.; Shaik, S.S.; Hiberty, P. C. J , Chem. SOC., Perkin Trans. 1992, 2, 1019. Sini, G.; Shaik, S. S.; Lefour, J.-M.; Ohanessian, G.; Hiberty, P. C. J . Phys. Chem. 1989, 93, 5661. ( 5 ) Shaik, S.; Ioffe, A.; Reddy, A. C.; Pross, A. J . Am. Chem. SOC. 1994, 116, 262. (6) Mulliken, R. S. J . Phys. Chem. 1952,56,801;J . Chim. Phys. 1964, 20, 20. Coulson, C. A. In Hydrogen Bonding; Hadii, D., Thompson, H. W., Eds.; Pergamon: London, 1959. Bratoi, S. Adv. Quantum Chem. 1967, 3,209. Ilczyszyn, M.; Ratajczak, H.; Skowronek, K. Magn. Reson. Chem. 1988,26,445. Hasegawa, M.; Daiyasu, K.; Yomosa, S. J. Phys. SOC.Jpn. 1970, 28, 275, 1304. Warshel, A.; Russel, S. J. J . Am. Chem. SOC.1986, 108, 6569. Reed, A. E.; Curtis, L. A,; Weinhold, F. A. Chem. Rev. 1988, 88, 899. (7) Some support for the Mulliken charge transfer picture-see refs 3c and 6-has been recently found for the HCl acid ionization in water [Ando, K.; Hynes, J. T. Molecular Mechanism of HCl Acid Ionization in Water: Ab Initio Potential Energy Surfaces and Monte Carlo Simulations. Submitted to J . Am. Chem. SOC.]. (8) Ross, A. Adv. Phys. Org. Chem. 1984, 106, 1227. (9) Hiberty, P. C.; Leforestier, C. J . Am. Chem. SOC.1978, 100, 2012. Hiberty, P. C.; Ohanessian, G. Ibid. 1982, 104, 66. Hiberty, P. C. Int. J . Quantum Chem. 1981, 19, 259. (10) Persico, M. In Spectral Line Shapes; Rostas, F., Ed.; de Gruyter: Berlin, 1985; Vol. 3, p 587. Ruedenberg, K.; Atchity, G. J. J . Chem. Phys. 1993, 99, 3799. Pacher, T.; Cederbaum, L. S.;Koppel, H. Adv. Chem. Phys. 1993, 84, 293. Cooper, D. L.; Gerratt, J.; Raimondi, M. Ibid. 1987, 67, 319. Bemardi, F.; Robb, M. A. Ibid. 1987, 67, 155. (11) Bianco, R.; Hynes, J. T. VB Resonance Theory in Solution. I. Multi-State Formulation; II. 12- * I I- in Acetonitrile. Submitted to J . Chem. Phys. (12) The Lowdin orthogonalization method preserves substantially the meaningful character of a set of nonorthogonal, diabatic VB states [Carlson, B. C.; Keller, J. M. Phys. Rev. 1957, 105, 1021. See, in particular, the last reference in ref 3a by Gertner et al. For other orthogonalization procedures, see the first reference in ref 3a. (13) This term is approximately' the ratio between the reorganization free energy factor associated with the electronic polarization---30 kcaU mol for e.g. typical electron transfers in polar solvents-and twice the energy difference between the ground and excited states of the electronic polarizatiorr-3-7 eV. (14) Atlas of Spectral Data and Physical Constants For Organic Compounds, 2nd ed.; Grasselli, J. G., Ritchey, W. M., Eds.; The Chemical Rubber Co.: Cleveland, 1975. (15) Kim, H. J.; Hynes, J. T. J . Chem. Phys. 1990, 93, 5194, 5211. (16) It can be easily shown for any number of states by taking the limit for {eo} m of ad obtained by inverting a,,-l (11). (17) Gehlen, J. N.; Chandler, D.; Kim, H. J.; Hynes, J. T. J . Phys. Chem. 1992, 96, 1748. (18) Basilevsky, M. V.; Chudinov, G. E.; Newton, M. D. Chem. Phys. 1994, 179, 263. (19) References 15 and 1 provide extensive references to earlier work in the BO and SC limits, largely in an equilibrium context. (20) Bianco, R.; Hynes, J. T. A Three Valence Bond State Treatment of the ClCH3C1 S N Reaction ~ in Solution. Manuscript in preparation.
+
This derivative (ignored in ref 1 for simplicity) has a linear structure in {c,} as required in connection with (AS), (A.4), and (32); its contribution in (AS) correctly reduces to the SC and BO limits.
References and Notes (1) Kim, H. J.; Hynes, J. T. J . Chem. Phys. 1992, 96, 5088. (2) Kharkats, Y. I.; Komyshev, A. A,; Vorotyntsev, M. A. J. Chem. Soc., Faraday Trans. 2 1976, 72, 361. (3) (a) Kim, H. J.; Hynes, J. T. J . Am. Chem. SOC.1992,114, 10508, 10528. Kim, H. J.; Bianco, R.; Gertner, B. J.; Hynes, J. T. J . Phys. Chem. 1993, 97, 1723. Mathis, J. R.; Kim, H. J.; Hynes, J. T. lbid. 1993, 115, 8248. Mathis, J. R.; Hynes, J. T. Ibid. 1994,98, 5445, 5460. Fonseca, T.; Kim, H. J.; Hynes, J. T. J. Photochem. Photobiol. A: Chem. 1994,82,67. Fonseca, T.; Kim, H. J.; Hynes, J. T. J . Mol. Liq. 1994, 60, 161. Gertner, B. J.; Ando, K.; Bianco, R.; Hynes, J. T. Chem. Phys. 1994, 183, 309. (b) Mathis, J. R.; Bianco, R.; Hynes, J. T. J . Mol. Liq. 1994,61, 8 1. (c) Juanos i Timoneda, J.; Hynes, J. T. J . Phys. Chem. 1991, 95, 10431.
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