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Slopes of the energy surface of transition metal ions in solution: A molecular orbital calculation
CHEMICAL PHYSICS LETTERS
Volume 173, number 1
28 September 1990
Slopes of the energy surface of transition metal ions in solution: a molecular orbital calculation Shaded U.M. Khan and Zheng Yu Zhou Department of Chemistry, Duquense University,Pittsburgh, PA 15282, USA Received 9 May 1990; in final form 26 June 1990
Application of the semiempirical quantum-chemical INDO/ZMO method in computing the slopes of the energy surface of various ions participating in electron-transfer reactions in solution is presented in this paper. The values of slopes obtained from INDO/Z-MO agree with those obtained from the classical improved-average-dipole-orientation (L4DO) method. Theoretical results of electronic-transmission coefficients of electron-transfer reactions obtained using values of slopes from INDO/Z-MO as well as IADO methods are found in close agreement with those obtained from experimental values of rate constants The values of transmission coefficient are found to be less than unity and these indicate the non-adiabatic nature of electron-transfer reactions involving the transition metal ions studied in this work.
1. Introdllction Recently, electron-transfer reactions involving transition metal ions in solution have been the focus of theoretical and experimental studies [l-5]. We shall consider the following class of reactions involving the hexa-coordinated complex in solution: ML;+‘+ML:=ML:+MG+’
,
(1)
where M is the transition metal and superscripts z and z+ 1 denote the ionic charges of the reactants in solution. The experimental values of rate constant, k,, for such electron-transfer reactions can be expressed as [6] %=&&exp(-AG’IRT),
(2)
where AGSLis the free energy of activation, Kelis the electronic transmission coefficient for outer-sphere electron-transfer reactions in solution and Z,, is an effective frequency for nuclear motion. According to the Landau and Zener formalism, the electronic transmission coeffkient, &, can be expressed as [ 6,7] 1FZ,=1-exp(-4~2Hi2fIhV,IS2-S11),
(3)
where S, is the slope of the energy surface in the .MLa’ and & is that in the ML:+. Hif is the tran-
sition matrix element between the two electronic states of El and E,, respectively (fig. 1), h is Planck’s constant, V, is the relative velocity of approach of ML:+ and ML:+ during reaction. Determination of the values of the transmission coefficient, K,, is essential to compute the correct values of the free energy of activation, AG*, from the experimental values of the rate constant, k, to generate the correct theoretical values of the rate constant and to determine the adiabatic&y and non-adiabaticity of the electron-transfer reaction in solutions. Recent studies [ 1 ] attempted the computation of the transition matrix and the transmission coefficient for electrontransfer reactions involving three transition metal ions in solution. However, in these studies [ 1,2] no attempt was made to calculate the individual slopes, 5, and S, of the energy surface. In this paper, we focus on computing the slopes of the energy surface using the INDO/ZMO as well as the classical improved-average-dipole-orientation (MD0 ) method.
2. INDOIZMO method The values of the slopes of energy surface, Si ( i= z or z+ I), can be obtained at the transition state from the relation,
0009-2614/90/S 03.50 Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
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Volume 173, number 1
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-W~~/WP=,LW~,
CHEMICAL PHYSICS LETTERS
(4)
where r, is the equilibrium ion-ligand bond distance and fAq represents approximately the distance between the equilibrium position of the ion-ligand bond and that at the transition state, r*. E(r) is the energy of the complex as a function of the ion-l&and bond distance, r, and can be obtained using the quantum-chemical INDO/ZMO method. The INDO/Z-MO formalism employed here is the one developed by Bacon and Zemer [ 81. The advantage of this INDO/ZMO method is that it was found quite useful in determining the energy of transition metal complexes [8-l 11. We also found [ 12J the INDO/ZMO method very useful in computing the inner-sphere reorganization energy for electrontransfer reactions in solution involving transition metal complexes. 2. I. The choice of parameters For INDO/ZMO calculation, we have used the basis sets 2s, 2p for N and 0 and 1s for the H atom. Basis sets 4s, 4p, 3d for the transition-metal ions, V2+, V3’, CP’, C?‘, Mn2+, Mn3+, Fe2+, Fe’+, Co2+ and Co3+, etc., involving aquo and ammine ligands were used. A basis set of single Slater-type orbitals (STO) has been characterized by the choice of the exponential constant, I&For hydrogen, a value of I+ 1.2 is taken [ 131. For the elements of N and 0, orbital exponents, t, were obtained from the work of Clementi and Raimondi [ 13). Though single orbitals are known to be somewhat inaccurate ford-orbitals, we would like to have the simplicity of “single” functions by adopting a multiple representation in the sense that we use values of e for different r values where r represents the ion-l&d distances. This greatly reduces the length of the computation. In addition, the orbital exponents were obtained from Bacon and Zemer [ 8 1. For the elements H, N and 0, we have used the one-center core UT integrals derived by Bacon and Zemer 181. For the transition metal ions, the onecenter core integrals vary with the atomic configurations and hence with the occupation number, n. As a result, for various valence states, we have therefore utilized the following equation as reported by Oleari, Sipio and Michaelis [ 141: 38
UJ=IJ+Aj-
28 September 1990
c niG:+$G$ (i
,
(5)
where ij, AjyG: and G$ are the ionization potential, electronegativity, the Coulomb and the exchange integrals, respectively. Zjand Aj values were taken from ref. [ 81. The semiempirical quantities G$ and G; are taken from the tabulated values given for the elements hydrogen through silver [ 14,lS 3. The relation of the bonding parameter, /JAB,between the non-diagonal matrix element, Hij and the overlap integral, S,, according to Pople’s approximation is Hij = B.&u >
(6)
where jIAB= f (PA+ /lB ) , and fiAand j$,, as the bonding parameters related to atom A and B, respectively, were taken from Bacon and Zerner [ 81. 2.2. Calculation of E(r) For the complexes of transition metal ions, the energy E(r) versus the ion-l&and bond distance r for a few distances was calculated by using the INDO/ 2-MO method. In order to get a smooth energy profile, E(r) as a function of r, the polynomial curvefitting method was used. Since second-power curve fitting was not sufficient, third-power curve fitting was used, i.e. E(r)=a+br+cr2+dr3,
(7)
where E(r) is the energy of the transition metal complex, r is the distance between the ion and a ligand and a, 6, c and dare the coefficients of the curve litting. The results of the energy-distance profiles are given in figs. 1 and 2 for the Fe3+ (H,0)6 and Fe? ( H20)6 systems, respectively. Similar results for other transition metal complexes were obtained but are not given for simplicity. From these figures, the equilibrium ion-l&and bond lengths, r,, of the oxidized and the reduced ions were obtained. Differentiating E( r) with respect to r, we obtained the slope of the energy surface at the transition state using eq. (4). The values of these slopes are given in table 1 (see columns 5 and 6).
Fe3+
E(r)
-
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Fig, 1. TheenergyE(r) versus theion-ligand distancerplotofthe [Fe(H,0),13+ system. The circles (0) areobtained fromthe INDO/ 2-MO calculation and the smooth line (-) is obtained using the curve-tilting program.
3. Improved-averaged&k-orientation (IADO) method The classical IADO method [ 16 ] is based on iondipole orbiting models [ 16-2 I]. The IADO method was found successful in calculating tbe inner-sphere reorganization energy for electron-transfer reactions involving transition metal complexes [ 19,201. The interaction energy due to orientations of a dipole (ligand) with the central ion is expressed as [ 16l8] - U(Z, r)=ze2a/2r~+zeD(cos
e>/r:,
and Bowers [ 17 1. For H20, (cos f3>= 0.80 and for NHj, (cos 8) =0.82 [20]. Differentiating - V(z, r) with respect to r, one obtains the slope of the energy profile as
=
2ze2a/P5+ 2zeD( cos Q /r*3 ,
(9)
where i represents z or z+ 1. The values of the slopes, S, and S, that are obtained from the IADO method for ions of charges z t 1 and z are given, respectively, in columns 8 and 9 in table 1.
(8)
where e is the electronic charge, (Yis the polarizability, D is the dipole moment of the ligand (dipole), r, is the ion-ligand bond distance with charge z and 0 is the angle between the dipole direction and r,. There are a number of ion-dipole orientation models [ 17-211 which allow an estimation of the average value of cos e, i.e. (cos e). The expression for { cos e> is originally due to Su
4. Values of transmission coefficient,& The value of the transmission coefficient, &, obtained from the experimental value of the rate constant, k, can be expressed as
(10) 39
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CHEMICAL PHYSICS LETTERS
Fe’+
-139
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(H20)6
-143
-145 I.L
1.9
1.b
I.6
2.0
2.2
2.4
2.6
Fig. 2. The energy E( r) versus the ion-ligand distance rplot of the [ Fe( Hz0 b6]‘+ system. The circles ( 0) are obtained from the INDo/ 2-MO calculation and the smooth line (-) is obtained using the curve-fitting program.
where k, is the experimental value of the rate constant, Z,, is the effective frequency of nuclear motion with the value of 10” dm3 mol-’ s-’ [ 6,2 11, and AG” is the value of the free energy of activation which can be expressed as AG”= $Rin t 4Rout = AGi” t AGO,,s
(11)
For electron-transfer reactions in solution, the values of the inner-sphere reorganization energy, Rinr can be obtained using the expression [ 63
Ri”=2n[f,f,+,l(f,tf,+,)lA4’>
(12)
where n is the total number of ligands in the inner shell of an ion, Aq = r, - rr+ , , and r, and r,, , are the equilibrium ion-ligand bond lengths (experimental) for the ionic charges z and z+ 1, respectively. fi andf,, I are the force constants of bonds between the metal ion (with charges z and zt 1, respectively), and its associated ligands can be obtained using the relation [ 19,201 40
A =4x2mLc20f/N,
(13)
where Of is the experimentally observed symmetric stretching frequency of the ith ion-ligand bond, c is the velocity of light, mL is the molecular weight of the ligand, and N is Avogadro’s number. Rout is obtained from the data of Rt,,ti from DeMay’s photoemission experiments [ 22,231 using the relation L
=J&ota~-&
-
(14)
For the cases when R,,, are not available, the values of R, are obtained from the following equation [21]: &,,=e2(1/~,P-l/e,) X [1/2r,+l/2r,,,-1/(r,+r,+,)15
(15)
where r, and rz+ Lare the experimental values of radii of two ions including the first solvation shell, copand es are the experimental values of the optical and static
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CHEMICAL. PHYSICS LETTERS
28 September 1990
Table l Slopes of the energy surface for transition metal ion-ligand ( Hz0 and NHs) bond in solution System
Rm”
Routb’
(MG+‘J+)
(W
(ev)
M
L
V Cr MIl Fe co Rll Cr Mll Fe co RU
Hz0 H20 Hz0 H2O H20
Hz0 NH3 NH3 NH3 NH3 NH3
2.68 2.42 3.08 1.98 2.12 0.28 2.26 1.76
1.22 2.71 0.42
0.79 0.85 0.57 1.16 1.19 1.15h’ 1.08 1.09 1.10 1.11 b, l.lOh’
INDO/ZMO ‘) ( 10’ erg/cm)
IADO d, (IO’ erg/cm)
SI
32
IS,-&I
Sl
s2
IS2-&l
-6.38 -7.45 -6.93 -7.01 -8.21
9.23 10.17 9.98 11.4 11.5
15.61 17.62 16.91 18.41 19.71
-7.3 -7.6 -8.0 -8.6
10.3 11.3 11.8 11.5
17.6 18.9 19.8 20.1
- 5.74 -6.13 -6.74 - 7.02 -7.37 -6.88 -6.21 -6.87 -7.15 -7.53 -7.15
8.61 9.19 10.12 10.52 11.06 10.32 9.56 10.6 11.0 11.6 11.0
14.4 15.3 16.9 17.5 18.4 17.2 15.8 17.5 18.2 19.1 18.2
‘) Data from ref. [20] and Aq values for V2+(H20)6/V3*(H10)6, C?+(H20),&r3+(H20)~, Mn2+(H20)6/Mn3f(HzO)s, Fti*(HzO)6/Fe3+(H20)s, CO~+(H~O)&O~+(H~O)~ and Ru~+(H~O)~/RI?+(H~O)~ systems were 0.19,0.18, 0.20, 0.16,0.17 and 0.08 A, respectively [ZO]. For C?+(NH&/Cr3+(NH&., Mn2+(NH,)6/Mn’+(~I)L, F?+(NH&/Fe’+(NH&, CO~+(NI-&/CO~+(NH~)~ and Ru*+(NH~)~/Ru~+(NH~)~ were 0.18,0.20,0.16,0.22 and 0.04 A, respectively [ZO]. ‘) Data obtained from experimental data [ 201. ‘) Data from eq. (4). ‘) Data from eq. (9). h, Data fromeq. ( 15).
dielectric constants for the solvent (cop= 1.777, ~,=78.3, for water as the solvent).
5. Results and discussion The slopes of the energy surfaces at the transition state for the aquo and ammine complexes of transition metal ions are given in table 1. It is observed that the results of the slopes S1 and S, obtained from the IND0/2-MO method and from the classical IADO method are in very good agreement for both aquo and ammine complexes (compare columns 5 and 6 with columns 8 and 9 in table t ). The values of IS, -& 1 obtained by the INDO/Z-MO method and the IADO method are also in very good agreement (compare column 7 with 10 in table 1). Theoretical values of the transmission coefficient, Kel, obtained by using the INDO/ZMO and the IADO method are given in table 2 (see columns 6 and 7). For the calculation of J& the values of the transition matrix, H#, were taken from recently reported results of German and Kuznetsov [ 5 ] (see column 4 in table 2). It is observed that the results of &I obtained using the INDO/ZMO dnd IADO methods are in close agreement.
These values of K,, are also in reasonable agreement with the values of the electronic transmission coefficient, &,, when obtained from experimental rate constant, k, data by including contributions of both inner-sphere and outer-sphere reorganization energies in eq. ( 10) (compare column 5 with columns 6 and 7 in table 2) except for the Fe2+(H20)6/Fe3+(H20)6 and V2+(H20)6/ V 3+ (H,O), redox systems. For these two systems, unrealistically high values of & are obtained if both AGi, and AG,, are taken into account in eq. ( 10). Reasonable values of K,, are obtained only if AGi, is taken into account in eq. ( 10 ) (see the entries in parentheses in column 5 in table 2). This indicates that for these complexes, AGi, is sufficient to take into account the free energy of activation. We did not calculate &, using the INDO/ZMO method for ruthenium complexes due to lack of availability of parameters for Ru. However, for ruthenium complexes good agreement is found between &, (obtained by using slopes of the IADO method) and Kd obtained from experimental values of the rate constants k, (compare column 7 with 5 in table 2). It is found that the slopes S, and S, obtained from the IND0/2-MO method are higher than those ob41
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Table 2 Values of transmission coefficients for outer-sphere electron-transfer reactions in solution System (ML:+“+) M
L
Rate constant k, (m-l s-i) (exp.)
V
Hz0
1.5 x10-2”
SO
Cr Fe
Hz0 Hz0
2 Xlo-Ss) 4.2 s’
120 50
Ru Cc Ru
Hz0 NH, NH3
1 x10”’
50 50 50
1 x10-‘” 3.2x103”
Hir ‘) (cm-‘)
%
b)
&
78.1 (3.41 x 10-Z) c) 1.48 x IO-’ 3.39x lo* (4.83x IO-‘) Cl 7.13x10-1 1.59x 10-Z 8.94x lo-’
INDO ”
IADod’
4.13x 10-Z
4.62x 1O-2
1.73x 10-I 3.66x 1O-2
2.13x 10-l 3.86x 1O-z
3.46x lO-2
5.23x lo-’ 3.64x lo-’ 4.95 X 10-s _
_ 8; Data from ref. [ Sj. b, Data from usiug& =k/[Z&exP( -AG”>&]. ‘1 Data from eq. (4) using IS,--& 1of INDO/Z-MO method. d, Data from eq, (4) using IS,-& ) of IADD method. (1 Datafromusing&=t/[&exp( -AG$/kT)], AGg=iRi,,. “Datafrom ref. [24]. s) Datafromrefs. [25,26]. h) Data from refs. [ 24,261. i) Data from refs. [ 27,281. j) Data from refs. [24,29]. k, Data from refs. [ 30,311.
tained from the classical IADO method for most cases. This observation may be attributed to the fact that the quantum-chemical INDO/Z-MO method takes into account the bonding contribution due to electron overlap between the ion and the ligands, whereas the classical IADO method takes into account only the electrostatic interactions. Consequently, the values of X; obtained from the INDO/ 2-MO method are smaller (closer to those obtained from experimental rate-constant data) than those obtained from the classical IADO method. Further, the closeness of results of the classical IADO method with those of the INDO/ZMO method may be due to the fact that the force field in the former also accounts for the short-range interaction in terms of a polarizability contribution instead of an electron overlap as in the INDO/Z-MO method. In conclusion, it is important to point out that the values of electronic transmission coefftcient are found to be less than unity (see table 2) and thus indicate that the outer-sphere electron-transfer reactions in solution involving aquo and ammine complexes of the few transition metal ions studied in this work are non-adiabatic in nature. Theoretical results of electronic transmission coefficient obtained using values of slopes from the quantum-chemical INDO/ZMO method are found to be in close agreement with those obtained from the classical IADO method and also 42
with those obtained from experimental values of rate constants k, (see table 2), and hence justifies the validity of the use of the latter. This further indicates that the IADO method based on classical ion-dipole orbiting models [ 16-191 gives rise to an adequate energy profile that can be utilized for these calculations.
References [l] M.D. Newton, J. Phys. Chem. 92 (1988) 3049. [2] G.E. McManis, A.K. Mishra and M.J. Weaver, J. Chem. Phys. 86 (1987) 5550. [ 31 S. Iarsson, K. Stahl and M.C. Zemer, Inorg. Chem. 25 (1986) 3033.
[ 41 B.S. Brunschwig and N. Sutin, Comments Inorg. Chem. 6 (1987) 209. [ 51 E.D. German and A.M. Kuznetsov, J. Chem. Sot. Faraday Trans.181 (1985) 1153. [6]J. O’M. Bockris and S.U,M. Khan, Quantum electrochemistry (Plenum Press, New York, 1979) ch. 13, p, 162. [7]C.Zener,Proc.Roy.Soc.A137(1932)696. [8] A.D. Bacon and M.C. Zemer, Theoret. Chim. Acta 53 (1979) 21. [9] J.A Rodriguez and C.T. Campbell, J. Phys. Chem. 9 1 (1987) 2161. [ 10 ] M.C. Zcmer, G.H. Loew, R.F. Krichner and UT. MuellerWesterhoff, J. Am. Chem. Sot. 102 ( 1980) 589.
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[ 111A. Waleh, ML. Cher, G.H. Loew and T. Mueller-Westerhoff, Theoret. Chim. Acta 65 ( 1984 ) 167. [ 121 Z.Y. Zhou and SUM. Khan, J. Phys. Chem. 93 (1989) 5292. [ 131 J.A. Pople and D.L. Beveridge, Approximate molecular orbital theory (McGraw-Hill, New York, 1970) ch. 3. [ 141 L. Oleari, L.D. Sipio and G.D. Michaelis, Mol. Phys. 10 (1966) 97. [ 15 ] LD. Sipio, E. Tondell, G.D. Michaelis and L. Oleari, Chem. Phys. Letters 11 ( 1971) 287. [ 161 T. Su and M.T. Bowers, J. Chem. Phys. 58 (1973) 3027. [ 171 L. Bass, T. Su, W.J. Chesnavich and M.T. Bowers, Chem. Phys. Letters 34 (1975) 119; W.J. Cbesnavich, T. Su and M.T. Bowers, J. Chem. Phys. 72 (1980) 2641. [ 181 D.R. Bates, Chem. Phys. Letters 82 ( 1981) 396. [ 191 M.S.TtumliandS.U.M.Rban, J. Chem.Soc. FaradayTram. 182 (1986) 2911.
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[20] M.S. Tunuli and S.U.M. Khan, 3. Phys. Chem. 91 (1987) 3474. [21] R.A. Marcus, Ann. Rev. Phys. Chem. 15 (1965) 155. [ 22 ] P. Delahay, Chem. Phys. Letters 87 ( 1982 ) 607. [ 23 ] P. Delahay and A Dxiedxic, J. Chem. Phys. 80 ( 1984) 5793. [ 241 B.S. Brunschwig, C. Creutx, D.H. Macartney, T.-K. Sham and N. Sutin, Faraday Discussions Chem. Sot. 74 (1982) 113. [25] I. Ruff and M. Zimonyi, Electrochim. Acta 18 (1973) 515. [26] J. Silverman and N.A. Bonner, J. Phys. Chem. 56 (1952) 846. [27] P. Bernhard, H.B. Burgi, H. Lehmann and A. Ludi, Inorg. Chem. 21 (1982) 3936. [28] W. Bottcher, G.M. Brown and N. Sutin, Inorg. Chem. 18 ( 1979) 1447. [29] A.M. Sargeson, Chem. Britain 15 (1979) 23. [30] T.J. Meyer and H. Taube, Inorg. Chem. 7 (1968) 2369. [3l]H.C.StynesandJ.A.Ibers, Inorg.Chem. 10 (1971) 2304.
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Volume 173, number 1
28 September 1990
Slopes of the energy surface of transition metal ions in solution: a molecular orbital calculation Shaded U.M. Khan and Zheng Yu Zhou Department of Chemistry, Duquense University,Pittsburgh, PA 15282, USA Received 9 May 1990; in final form 26 June 1990
Application of the semiempirical quantum-chemical INDO/ZMO method in computing the slopes of the energy surface of various ions participating in electron-transfer reactions in solution is presented in this paper. The values of slopes obtained from INDO/Z-MO agree with those obtained from the classical improved-average-dipole-orientation (L4DO) method. Theoretical results of electronic-transmission coefficients of electron-transfer reactions obtained using values of slopes from INDO/Z-MO as well as IADO methods are found in close agreement with those obtained from experimental values of rate constants The values of transmission coefficient are found to be less than unity and these indicate the non-adiabatic nature of electron-transfer reactions involving the transition metal ions studied in this work.
1. Introdllction Recently, electron-transfer reactions involving transition metal ions in solution have been the focus of theoretical and experimental studies [l-5]. We shall consider the following class of reactions involving the hexa-coordinated complex in solution: ML;+‘+ML:=ML:+MG+’
,
(1)
where M is the transition metal and superscripts z and z+ 1 denote the ionic charges of the reactants in solution. The experimental values of rate constant, k,, for such electron-transfer reactions can be expressed as [6] %=&&exp(-AG’IRT),
(2)
where AGSLis the free energy of activation, Kelis the electronic transmission coefficient for outer-sphere electron-transfer reactions in solution and Z,, is an effective frequency for nuclear motion. According to the Landau and Zener formalism, the electronic transmission coeffkient, &, can be expressed as [ 6,7] 1FZ,=1-exp(-4~2Hi2fIhV,IS2-S11),
(3)
where S, is the slope of the energy surface in the .MLa’ and & is that in the ML:+. Hif is the tran-
sition matrix element between the two electronic states of El and E,, respectively (fig. 1), h is Planck’s constant, V, is the relative velocity of approach of ML:+ and ML:+ during reaction. Determination of the values of the transmission coefficient, K,, is essential to compute the correct values of the free energy of activation, AG*, from the experimental values of the rate constant, k, to generate the correct theoretical values of the rate constant and to determine the adiabatic&y and non-adiabaticity of the electron-transfer reaction in solutions. Recent studies [ 1 ] attempted the computation of the transition matrix and the transmission coefficient for electrontransfer reactions involving three transition metal ions in solution. However, in these studies [ 1,2] no attempt was made to calculate the individual slopes, 5, and S, of the energy surface. In this paper, we focus on computing the slopes of the energy surface using the INDO/ZMO as well as the classical improved-average-dipole-orientation (MD0 ) method.
2. INDOIZMO method The values of the slopes of energy surface, Si ( i= z or z+ I), can be obtained at the transition state from the relation,
0009-2614/90/S 03.50 Q 1990 - Elsevier Science Publishers B.V. (North-Holland)
37
Volume 173, number 1
4=
-W~~/WP=,LW~,
CHEMICAL PHYSICS LETTERS
(4)
where r, is the equilibrium ion-ligand bond distance and fAq represents approximately the distance between the equilibrium position of the ion-ligand bond and that at the transition state, r*. E(r) is the energy of the complex as a function of the ion-l&and bond distance, r, and can be obtained using the quantum-chemical INDO/ZMO method. The INDO/Z-MO formalism employed here is the one developed by Bacon and Zemer [ 81. The advantage of this INDO/ZMO method is that it was found quite useful in determining the energy of transition metal complexes [8-l 11. We also found [ 12J the INDO/ZMO method very useful in computing the inner-sphere reorganization energy for electrontransfer reactions in solution involving transition metal complexes. 2. I. The choice of parameters For INDO/ZMO calculation, we have used the basis sets 2s, 2p for N and 0 and 1s for the H atom. Basis sets 4s, 4p, 3d for the transition-metal ions, V2+, V3’, CP’, C?‘, Mn2+, Mn3+, Fe2+, Fe’+, Co2+ and Co3+, etc., involving aquo and ammine ligands were used. A basis set of single Slater-type orbitals (STO) has been characterized by the choice of the exponential constant, I&For hydrogen, a value of I+ 1.2 is taken [ 131. For the elements of N and 0, orbital exponents, t, were obtained from the work of Clementi and Raimondi [ 13). Though single orbitals are known to be somewhat inaccurate ford-orbitals, we would like to have the simplicity of “single” functions by adopting a multiple representation in the sense that we use values of e for different r values where r represents the ion-l&d distances. This greatly reduces the length of the computation. In addition, the orbital exponents were obtained from Bacon and Zemer [ 8 1. For the elements H, N and 0, we have used the one-center core UT integrals derived by Bacon and Zemer 181. For the transition metal ions, the onecenter core integrals vary with the atomic configurations and hence with the occupation number, n. As a result, for various valence states, we have therefore utilized the following equation as reported by Oleari, Sipio and Michaelis [ 141: 38
UJ=IJ+Aj-
28 September 1990
c niG:+$G$ (i
,
(5)
where ij, AjyG: and G$ are the ionization potential, electronegativity, the Coulomb and the exchange integrals, respectively. Zjand Aj values were taken from ref. [ 81. The semiempirical quantities G$ and G; are taken from the tabulated values given for the elements hydrogen through silver [ 14,lS 3. The relation of the bonding parameter, /JAB,between the non-diagonal matrix element, Hij and the overlap integral, S,, according to Pople’s approximation is Hij = B.&u >
(6)
where jIAB= f (PA+ /lB ) , and fiAand j$,, as the bonding parameters related to atom A and B, respectively, were taken from Bacon and Zerner [ 81. 2.2. Calculation of E(r) For the complexes of transition metal ions, the energy E(r) versus the ion-l&and bond distance r for a few distances was calculated by using the INDO/ 2-MO method. In order to get a smooth energy profile, E(r) as a function of r, the polynomial curvefitting method was used. Since second-power curve fitting was not sufficient, third-power curve fitting was used, i.e. E(r)=a+br+cr2+dr3,
(7)
where E(r) is the energy of the transition metal complex, r is the distance between the ion and a ligand and a, 6, c and dare the coefficients of the curve litting. The results of the energy-distance profiles are given in figs. 1 and 2 for the Fe3+ (H,0)6 and Fe? ( H20)6 systems, respectively. Similar results for other transition metal complexes were obtained but are not given for simplicity. From these figures, the equilibrium ion-l&and bond lengths, r,, of the oxidized and the reduced ions were obtained. Differentiating E( r) with respect to r, we obtained the slope of the energy surface at the transition state using eq. (4). The values of these slopes are given in table 1 (see columns 5 and 6).
Fe3+
E(r)
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1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
Fig, 1. TheenergyE(r) versus theion-ligand distancerplotofthe [Fe(H,0),13+ system. The circles (0) areobtained fromthe INDO/ 2-MO calculation and the smooth line (-) is obtained using the curve-tilting program.
3. Improved-averaged&k-orientation (IADO) method The classical IADO method [ 16 ] is based on iondipole orbiting models [ 16-2 I]. The IADO method was found successful in calculating tbe inner-sphere reorganization energy for electron-transfer reactions involving transition metal complexes [ 19,201. The interaction energy due to orientations of a dipole (ligand) with the central ion is expressed as [ 16l8] - U(Z, r)=ze2a/2r~+zeD(cos
e>/r:,
and Bowers [ 17 1. For H20, (cos f3>= 0.80 and for NHj, (cos 8) =0.82 [20]. Differentiating - V(z, r) with respect to r, one obtains the slope of the energy profile as
=
2ze2a/P5+ 2zeD( cos Q /r*3 ,
(9)
where i represents z or z+ 1. The values of the slopes, S, and S, that are obtained from the IADO method for ions of charges z t 1 and z are given, respectively, in columns 8 and 9 in table 1.
(8)
where e is the electronic charge, (Yis the polarizability, D is the dipole moment of the ligand (dipole), r, is the ion-ligand bond distance with charge z and 0 is the angle between the dipole direction and r,. There are a number of ion-dipole orientation models [ 17-211 which allow an estimation of the average value of cos e, i.e. (cos e). The expression for { cos e> is originally due to Su
4. Values of transmission coefficient,& The value of the transmission coefficient, &, obtained from the experimental value of the rate constant, k, can be expressed as
(10) 39
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Fe’+
-139
28 September 1990
(H20)6
-143
-145 I.L
1.9
1.b
I.6
2.0
2.2
2.4
2.6
Fig. 2. The energy E( r) versus the ion-ligand distance rplot of the [ Fe( Hz0 b6]‘+ system. The circles ( 0) are obtained from the INDo/ 2-MO calculation and the smooth line (-) is obtained using the curve-fitting program.
where k, is the experimental value of the rate constant, Z,, is the effective frequency of nuclear motion with the value of 10” dm3 mol-’ s-’ [ 6,2 11, and AG” is the value of the free energy of activation which can be expressed as AG”= $Rin t 4Rout = AGi” t AGO,,s
(11)
For electron-transfer reactions in solution, the values of the inner-sphere reorganization energy, Rinr can be obtained using the expression [ 63
Ri”=2n[f,f,+,l(f,tf,+,)lA4’>
(12)
where n is the total number of ligands in the inner shell of an ion, Aq = r, - rr+ , , and r, and r,, , are the equilibrium ion-ligand bond lengths (experimental) for the ionic charges z and z+ 1, respectively. fi andf,, I are the force constants of bonds between the metal ion (with charges z and zt 1, respectively), and its associated ligands can be obtained using the relation [ 19,201 40
A =4x2mLc20f/N,
(13)
where Of is the experimentally observed symmetric stretching frequency of the ith ion-ligand bond, c is the velocity of light, mL is the molecular weight of the ligand, and N is Avogadro’s number. Rout is obtained from the data of Rt,,ti from DeMay’s photoemission experiments [ 22,231 using the relation L
=J&ota~-&
-
(14)
For the cases when R,,, are not available, the values of R, are obtained from the following equation [21]: &,,=e2(1/~,P-l/e,) X [1/2r,+l/2r,,,-1/(r,+r,+,)15
(15)
where r, and rz+ Lare the experimental values of radii of two ions including the first solvation shell, copand es are the experimental values of the optical and static
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Table l Slopes of the energy surface for transition metal ion-ligand ( Hz0 and NHs) bond in solution System
Rm”
Routb’
(MG+‘J+)
(W
(ev)
M
L
V Cr MIl Fe co Rll Cr Mll Fe co RU
Hz0 H20 Hz0 H2O H20
Hz0 NH3 NH3 NH3 NH3 NH3
2.68 2.42 3.08 1.98 2.12 0.28 2.26 1.76
1.22 2.71 0.42
0.79 0.85 0.57 1.16 1.19 1.15h’ 1.08 1.09 1.10 1.11 b, l.lOh’
INDO/ZMO ‘) ( 10’ erg/cm)
IADO d, (IO’ erg/cm)
SI
32
IS,-&I
Sl
s2
IS2-&l
-6.38 -7.45 -6.93 -7.01 -8.21
9.23 10.17 9.98 11.4 11.5
15.61 17.62 16.91 18.41 19.71
-7.3 -7.6 -8.0 -8.6
10.3 11.3 11.8 11.5
17.6 18.9 19.8 20.1
- 5.74 -6.13 -6.74 - 7.02 -7.37 -6.88 -6.21 -6.87 -7.15 -7.53 -7.15
8.61 9.19 10.12 10.52 11.06 10.32 9.56 10.6 11.0 11.6 11.0
14.4 15.3 16.9 17.5 18.4 17.2 15.8 17.5 18.2 19.1 18.2
‘) Data from ref. [20] and Aq values for V2+(H20)6/V3*(H10)6, C?+(H20),&r3+(H20)~, Mn2+(H20)6/Mn3f(HzO)s, Fti*(HzO)6/Fe3+(H20)s, CO~+(H~O)&O~+(H~O)~ and Ru~+(H~O)~/RI?+(H~O)~ systems were 0.19,0.18, 0.20, 0.16,0.17 and 0.08 A, respectively [ZO]. For C?+(NH&/Cr3+(NH&., Mn2+(NH,)6/Mn’+(~I)L, F?+(NH&/Fe’+(NH&, CO~+(NI-&/CO~+(NH~)~ and Ru*+(NH~)~/Ru~+(NH~)~ were 0.18,0.20,0.16,0.22 and 0.04 A, respectively [ZO]. ‘) Data obtained from experimental data [ 201. ‘) Data from eq. (4). ‘) Data from eq. (9). h, Data fromeq. ( 15).
dielectric constants for the solvent (cop= 1.777, ~,=78.3, for water as the solvent).
5. Results and discussion The slopes of the energy surfaces at the transition state for the aquo and ammine complexes of transition metal ions are given in table 1. It is observed that the results of the slopes S1 and S, obtained from the IND0/2-MO method and from the classical IADO method are in very good agreement for both aquo and ammine complexes (compare columns 5 and 6 with columns 8 and 9 in table t ). The values of IS, -& 1 obtained by the INDO/Z-MO method and the IADO method are also in very good agreement (compare column 7 with 10 in table 1). Theoretical values of the transmission coefficient, Kel, obtained by using the INDO/ZMO and the IADO method are given in table 2 (see columns 6 and 7). For the calculation of J& the values of the transition matrix, H#, were taken from recently reported results of German and Kuznetsov [ 5 ] (see column 4 in table 2). It is observed that the results of &I obtained using the INDO/ZMO dnd IADO methods are in close agreement.
These values of K,, are also in reasonable agreement with the values of the electronic transmission coefficient, &,, when obtained from experimental rate constant, k, data by including contributions of both inner-sphere and outer-sphere reorganization energies in eq. ( 10) (compare column 5 with columns 6 and 7 in table 2) except for the Fe2+(H20)6/Fe3+(H20)6 and V2+(H20)6/ V 3+ (H,O), redox systems. For these two systems, unrealistically high values of & are obtained if both AGi, and AG,, are taken into account in eq. ( 10). Reasonable values of K,, are obtained only if AGi, is taken into account in eq. ( 10 ) (see the entries in parentheses in column 5 in table 2). This indicates that for these complexes, AGi, is sufficient to take into account the free energy of activation. We did not calculate &, using the INDO/ZMO method for ruthenium complexes due to lack of availability of parameters for Ru. However, for ruthenium complexes good agreement is found between &, (obtained by using slopes of the IADO method) and Kd obtained from experimental values of the rate constants k, (compare column 7 with 5 in table 2). It is found that the slopes S, and S, obtained from the IND0/2-MO method are higher than those ob41
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Table 2 Values of transmission coefficients for outer-sphere electron-transfer reactions in solution System (ML:+“+) M
L
Rate constant k, (m-l s-i) (exp.)
V
Hz0
1.5 x10-2”
SO
Cr Fe
Hz0 Hz0
2 Xlo-Ss) 4.2 s’
120 50
Ru Cc Ru
Hz0 NH, NH3
1 x10”’
50 50 50
1 x10-‘” 3.2x103”
Hir ‘) (cm-‘)
%
b)
&
78.1 (3.41 x 10-Z) c) 1.48 x IO-’ 3.39x lo* (4.83x IO-‘) Cl 7.13x10-1 1.59x 10-Z 8.94x lo-’
INDO ”
IADod’
4.13x 10-Z
4.62x 1O-2
1.73x 10-I 3.66x 1O-2
2.13x 10-l 3.86x 1O-z
3.46x lO-2
5.23x lo-’ 3.64x lo-’ 4.95 X 10-s _
_ 8; Data from ref. [ Sj. b, Data from usiug& =k/[Z&exP( -AG”>&]. ‘1 Data from eq. (4) using IS,--& 1of INDO/Z-MO method. d, Data from eq, (4) using IS,-& ) of IADD method. (1 Datafromusing&=t/[&exp( -AG$/kT)], AGg=iRi,,. “Datafrom ref. [24]. s) Datafromrefs. [25,26]. h) Data from refs. [ 24,261. i) Data from refs. [ 27,281. j) Data from refs. [24,29]. k, Data from refs. [ 30,311.
tained from the classical IADO method for most cases. This observation may be attributed to the fact that the quantum-chemical INDO/Z-MO method takes into account the bonding contribution due to electron overlap between the ion and the ligands, whereas the classical IADO method takes into account only the electrostatic interactions. Consequently, the values of X; obtained from the INDO/ 2-MO method are smaller (closer to those obtained from experimental rate-constant data) than those obtained from the classical IADO method. Further, the closeness of results of the classical IADO method with those of the INDO/ZMO method may be due to the fact that the force field in the former also accounts for the short-range interaction in terms of a polarizability contribution instead of an electron overlap as in the INDO/Z-MO method. In conclusion, it is important to point out that the values of electronic transmission coefftcient are found to be less than unity (see table 2) and thus indicate that the outer-sphere electron-transfer reactions in solution involving aquo and ammine complexes of the few transition metal ions studied in this work are non-adiabatic in nature. Theoretical results of electronic transmission coefficient obtained using values of slopes from the quantum-chemical INDO/ZMO method are found to be in close agreement with those obtained from the classical IADO method and also 42
with those obtained from experimental values of rate constants k, (see table 2), and hence justifies the validity of the use of the latter. This further indicates that the IADO method based on classical ion-dipole orbiting models [ 16-191 gives rise to an adequate energy profile that can be utilized for these calculations.
References [l] M.D. Newton, J. Phys. Chem. 92 (1988) 3049. [2] G.E. McManis, A.K. Mishra and M.J. Weaver, J. Chem. Phys. 86 (1987) 5550. [ 31 S. Iarsson, K. Stahl and M.C. Zemer, Inorg. Chem. 25 (1986) 3033.
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[ 111A. Waleh, ML. Cher, G.H. Loew and T. Mueller-Westerhoff, Theoret. Chim. Acta 65 ( 1984 ) 167. [ 121 Z.Y. Zhou and SUM. Khan, J. Phys. Chem. 93 (1989) 5292. [ 131 J.A. Pople and D.L. Beveridge, Approximate molecular orbital theory (McGraw-Hill, New York, 1970) ch. 3. [ 141 L. Oleari, L.D. Sipio and G.D. Michaelis, Mol. Phys. 10 (1966) 97. [ 15 ] LD. Sipio, E. Tondell, G.D. Michaelis and L. Oleari, Chem. Phys. Letters 11 ( 1971) 287. [ 161 T. Su and M.T. Bowers, J. Chem. Phys. 58 (1973) 3027. [ 171 L. Bass, T. Su, W.J. Chesnavich and M.T. Bowers, Chem. Phys. Letters 34 (1975) 119; W.J. Cbesnavich, T. Su and M.T. Bowers, J. Chem. Phys. 72 (1980) 2641. [ 181 D.R. Bates, Chem. Phys. Letters 82 ( 1981) 396. [ 191 M.S.TtumliandS.U.M.Rban, J. Chem.Soc. FaradayTram. 182 (1986) 2911.
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