Chemical Reviews REVIEW Table 1.Bond Dissociation Energies Calculated with Valence Table 2.Barriers for the Hydrogen Exchange Reactions X'+ Bond Methods180 HX-XH+X'(X=CH3,SiH3,GeH3,SnH3,PbH3,and H) De (kcal/mol) molecule HF CCSD VBSCE BOVB VBCISD VBCIPT bond basis set BOVB VBCISD CCSD(T) 35.1 26.5 33.0 23.1 25.8 25.5 exptl CH, SiH, 25.2 19.3 25.5 19.1 19.7 19.0 F-F 6-31G 362 32.3 32.8 GeH, 22.0 16.6 25.5 18.0 18.1 17.0 cc-pVTZ 37.9 36.1 34.8 38.3 SnH," 185 13.5 20.5 14.9 153 14.1 CI-CI 6-31G 40.0 41.6 40.5 PbHs 152 13.0 173 12.3 12.5 11.5 cc-pVTZ 50.0 56.1 52.1 58.0 H 9.84 20.6 10.2 10.0 Br-Br 6-31G 413 44.1 41.2 cc-pVTZ Energies in kilocalories per molebasis set:co5ref 44.0 50.0 48.0 45.9 117;columns 6 and 7,ref 116.Aug-cc-pVTZ basis set. “CCSD(T)】 F-Cl 6-31G 47.9 49.3 50.2 calculation. ce-pVTZ 53.6 58.8 55.0 60.2 H-H 6-31G 105.4 105.4 105.9 109.6 Li-Li 6-31G 20.9 212 21.1 24.4 R H:C-H 6-31G" 105.7 113.6 109.9 112.3 H:C-CH: 6-31G 94.7 90.0 95.6 96.7 HO-OH 6-31G 50.8 49.8 48.1 53.9 G H2N-NH2 6-31G 685 70.5 66.5 75.4±3 H:Si-H 6-31G* 93.6 90.2 91.8 97.6土3 △E HSi-F 6-31G 140.4 151.1 142.6 160±7 4 H3Si-Cl 6-31G 102.1 1012 98.1 113.7土4 "With Davidson correction.Two-structure calculations(H3SiF'is Reaction Coordinate omitted). Figure 5.VBSCD for a general reaction R-P.R and Pare the ground states of the reactants and products,and R*and p*are the promoted Thus,on the basis of the energies of the two diabatic states and excited states. the adiabatic state at the same level,one can derive the structural weights of the diabatic states in the ground state,the coupling This is illustrated in Table 2,where some barriers for hydrogen energy between the two diabatic states,and the transition energy abstraction reactions are calculated with various methods using to the first excited state without introducing any empirical the 6-31G*basis set (entries 1-5).It is seen that the BOVB, parameter. VBCISD,and VBCIPT methods provide barriers that are on par Alongside the VB/MM method due to Shurki et al.,MOVB/ with CCSD results using the same basis set.Moreover,for the MM is another kind of combined VB-type QM/MM method H"+H-H'-H-H+H'reaction (last entry),BOVB and which is based on the BLW method discussed above. VBCISD also provide a barrier value very close to the exact barrier and comparable to the experimental value of 9.8+0.2 3.APPLICATIONS kcal/mol.On the other hand,the less elaborate VBSCF method, This section covers a variety of applications of the VB methods which lacks dynamic correlation,generates barriers that are way too high in all cases. described above.It begins with a short survey of the accuracy of the methods and then discusses a panoramic set of applications. The ab initio VB method has also been applied to the excited states of several diatomic molecules,such as O2 NF,B2,and 3.1.Accuracy of Modern VB Methods LiB143s5181 All the VB studies provide satisfactory numerical Any computational method dealing with reactivity must results with intuitive insights for the bonding of excited states. describe as accurately as possible the elementary events of a Further examples,where VB-calculated barriers are on par reaction,i.e,bond breakage or bond formation.This feature has with benchmark values and close to experimental estimates,are been tested for various VB methods,and the shown in the following sections. emerging conclusion is that the methods which account for dynamic correlation,e.g.,BOVB,VBCI,and VBPT2,are reason- 3.2.Chemical Reactivity ably accurate.On the other hand,the VB methods that involve There are two fundamental questions that any model of only nondynamical correlation,e.g.GVB and VBSCF,generally chemical reactivity would have to answer:What are the origins provide good qualitative trends,but poor quantitative accuracy. of the barriers?What are the factors that determine the reaction Table 1 shows bonding energies for a sample of homopolar mechanisms?Since chemical reactivity involves bond breaking and heteropolar bonds calculated by various methods with basis and making,VB theory,with its focus on the bond as the key sets ranging from 6-31G*to cc-pVTZ.It is seen that BOVB constituent of the wave function,is able to provide a lucid model and VBCI give results on par with those of CCSD(T)calcula- that answers these two questions in a unified manner.The tions performed with the same basis sets.Furthermore,with a centerpiece of the VB model is the VB state correlation diagram large enough basis set,all these methods give results close to (VBSCD),which traces the energy of the VB configurations experimental values. along the reaction coordinate and provides a mechanism for Another important feature of modern VB methods is their the barrier formation and generation of a transition state in an ability to calculate reaction barriers with reasonable accuracy. elementary reaction (Figure 5).182 dx.dol.org/10.1021/cr100228r Chem.Rev.XXXX,XXX,000-000
K dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW Thus, on the basis of the energies of the two diabatic states and the adiabatic state at the same level, one can derive the structural weights of the diabatic states in the ground state, the coupling energy between the two diabatic states, and the transition energy to the first excited state without introducing any empirical parameter. Alongside the VB/MM method due to Shurki et al., MOVB/ MM177 is another kind of combined VB-type QM/MM method which is based on the BLW method discussed above. 3. APPLICATIONS This section covers a variety of applications of the VB methods described above. It begins with a short survey of the accuracy of the methods and then discusses a panoramic set of applications. 3.1. Accuracy of Modern VB Methods Any computational method dealing with reactivity must describe as accurately as possible the elementary events of a reaction, i.e., bond breakage or bond formation. This feature has been tested for various VB methods,113118,145,180 and the emerging conclusion is that the methods which account for dynamic correlation, e.g., BOVB, VBCI, and VBPT2, are reasonably accurate. On the other hand, the VB methods that involve only nondynamical correlation, e.g., GVB and VBSCF, generally provide good qualitative trends, but poor quantitative accuracy. Table 1 shows bonding energies for a sample of homopolar and heteropolar bonds calculated by various methods with basis sets ranging from 6-31G* to cc-pVTZ.180 It is seen that BOVB and VBCI give results on par with those of CCSD(T) calculations performed with the same basis sets. Furthermore, with a large enough basis set, all these methods give results close to experimental values. Another important feature of modern VB methods is their ability to calculate reaction barriers with reasonable accuracy. This is illustrated in Table 2, where some barriers for hydrogen abstraction reactions are calculated with various methods using the 6-31G* basis set (entries 15). It is seen that the BOVB, VBCISD, and VBCIPT methods provide barriers that are on par with CCSD results using the same basis set. Moreover, for the H• + HH0 f HH+H0• reaction (last entry), BOVB and VBCISD also provide a barrier value very close to the exact barrier and comparable to the experimental value of 9.8 ( 0.2 kcal/mol. On the other hand, the less elaborate VBSCF method, which lacks dynamic correlation, generates barriers that are way too high in all cases. The ab initio VB method has also been applied to the excited states of several diatomic molecules, such as O2, NF, B2, and LiB+ . 143,155,181 All the VB studies provide satisfactory numerical results with intuitive insights for the bonding of excited states. Further examples, where VB-calculated barriers are on par with benchmark values and close to experimental estimates, are shown in the following sections. 3.2. Chemical Reactivity There are two fundamental questions that any model of chemical reactivity would have to answer: What are the origins of the barriers? What are the factors that determine the reaction mechanisms? Since chemical reactivity involves bond breaking and making, VB theory, with its focus on the bond as the key constituent of the wave function, is able to provide a lucid model that answers these two questions in a unified manner. The centerpiece of the VB model is the VB state correlation diagram (VBSCD), which traces the energy of the VB configurations along the reaction coordinate and provides a mechanism for the barrier formation and generation of a transition state in an elementary reaction (Figure 5).182 Table 1. Bond Dissociation Energies Calculated with Valence Bond Methods180 De (kcal/mol) bond basis set BOVB VBCISDa CCSD(T) exptl FF 6-31G* 36.2 32.3 32.8 cc-pVTZ 37.9 36.1 34.8 38.3 ClCl 6-31G* 40.0 41.6 40.5 cc-pVTZ 50.0 56.1 52.1 58.0 BrBr 6-31G* 41.3 44.1 41.2 cc-pVTZ 44.0 50.0 48.0 45.9 FCl 6-31G* 47.9 49.3 50.2 cc-pVTZ 53.6 58.8 55.0 60.2 HH 6-31G** 105.4 105.4 105.9 109.6 LiLi 6-31G* 20.9 21.2 21.1 24.4 H3CH 6-31G** 105.7 113.6 109.9 112.3 H3CCH3 6-31G* 94.7 90.0 95.6 96.7 HOOH 6-31G* 50.8 49.8 48.1 53.9 H2N-NH2 6-31G* 68.5 70.5 66.5 75.4 ( 3 H3SiH 6-31G** 93.6 90.2 91.8 97.6 ( 3 H3SiF 6-31G* 140.4b 151.1 142.6 160 ( 7 H3SiCl 6-31G* 102.1 101.2 98.1 113.7 ( 4 a With Davidson correction.116 b Two-structure calculations (H3SiF+ is omitted). Table 2. Barriers for the Hydrogen Exchange Reactions X• + HX f XH + X0• (X = CH3, SiH3, GeH3, SnH3, PbH3, and H)a molecule HF CCSD VBSCF BOVB VBCISD VBCIPT CH3 b 35.1 26.5 33.0 23.1 25.8 25.5 SiH3 b 25.2 19.3 25.5 19.1 19.7 19.0 GeH3 b 22.0 16.6 25.5 18.0 18.1 17.0 SnH3 b 18.5 13.5 20.5 14.9 15.3 14.1 PbH3 b 15.2 13.0 17.3 12.3 12.5 11.5 Hc 9.8d 20.6 10.2 10.0 a Energies in kilocalories per mole. b 6-31G* basis set: columns 25, ref 117; columns 6 and 7, ref 116. c Aug-cc-pVTZ basis set.145 dCCSD(T) calculation. Figure 5. VBSCD for a general reaction R f P. R and P are the ground states of the reactants and products, and R* and P* are the promoted excited states
Chemical Reviews REVIEW This diagram applies to elementary reactions wherein the Practically,P,is a linear combination of covalent and ionic forms barrier can be described as the interplay of two major VB states, that contribute to the Lewis structure"X"+H-Y",as shown in that of the reactants and that of the products.It displays the the following: ground-state energy profile for the reacting system(bold curve), as well as the energy profiles for individual VB states (thinner Ψ.=C(X1+H-Y)+C2(X+HY)） curves);these latter curves are also called "diabatic"curves,while the full state energy curve (in bold)is called"adiabatic".Thus, +C3(X1+H-Y+) (46) starting from the reactant geometry on the left,the VB structure This combination is maintained in from R to p*throughout V.that represents the reactant's electronic state,R,has the lowest energy and merges with the ground state.Then,as one the reaction coordinate,while the coefficients of the contributing deforms the reacting molecules toward the product geometry, structures change and adapt themselves to the geometric change P.gradually rises and finally reaches an excited state p*that (e.g.,at infinite H---Y distance,C=1).The curve p,which stretches between P and R*is defined in an analogous manner. represents the VB structure of the reactants in the product geometry.A similar diabatic curve can be traced from P,the VB structure of Two definitions are possible for the diabatic state curves and the products in its optimal geometry,to R",the same VB struc- .In the variational diabatic configuration (VDC)method,the ture but in the reactant geometry.Consequently,the two curves energies of the diabatic states are variationally minimized,as is cross somewhere in the middle of the diagram.The crossing is of done in the applications described below unless otherwise noted. course avoided in the adiabatic ground state,owing to the mixing In this way,each diabatic state has the best possible combination of the two VB structures,which stabilizes the resulting transi- of coefficients and orbitals for this specific state.Alternatively,in tion state by a resonance energy term,labeled B.The barrier is the consistent diabatic configuration (CDC)method,18s the thus interpreted as arising from avoided crossing between two diabatic states are simply extracted from the ground-state wave diabatic curves which represent the energy profiles of the VB function by projection.As explained before,Ts the inconveni- ence of the CDC technique is that the so-constructed diabatic state curves of the reactants and products. The nature of the R*and p*promoted states depends on the states involve orbitals and VB coefficients of the ground state and reaction type and will be explained below using specific examples. are therefore not optimal for the diabatic states.By comparison, In all cases,the promoted state R*is the electronic image of P in the VDC technique gives quasi-variational quantities for all the the geometry of R,while p*is the image of R at the geometry of P. parameters in the diagram (f G,B). The G terms are the corresponding promotion energy gaps,B is Since the promoted state R*is the VB structure of P in the the resonance energy of the transition state,AE is the energy geometry of R,its electronic state is illustrated by barrier,and AEp is the reaction energy.The simplest expression R*=(X+F)-Y (47) for the barrier is given by where the H-Y bond is infinitely long,while the x"radical △E=fG:-B (42) (spin-up)experiences some Pauli repulsion with the electron of H,which is 50%spin-up and 50%spin-down.As such,the R* Here,the term fG,is the height of the crossing point,expressed as state is 75%a triplet state,and hence,the G gap is (3/4)AEsT, some fraction(f)of the promotion gap at the reactant side(G). AEsr being the singlet-triplet excitation of the X-H bond that A more explicit expression is undergoes activation. In the study of identity reactions(X=Y),it has been shown △E≈6G0+(Gp/2G0)△EP+(1/2G0)△EP2-B that,indeed,the promotion energy G.required to go from Go=0.5(G+G) 6=丘+6 43) he systematic VB ab initio calculations by the VBCI method have shown that,to a good approximation,G.can be expressed as which considers the two promotion gaps and f factors through their average quantities,Go and fo Equation can be further follows:189 simplified by neglecting the quadratic term and taking G/2Go as G:≈2D(X-H) (48】 1/2,thus leading to Moreover,a semiempirical derivation showed's9 that the reso- △E=f6G0+0.5△Em-B nance energy B is also proportional to D(X-H)by the following (44) expression: Equation 44 expresses the barrier as a balance of the contribu- B≈0.5D(X-H) (49) tions of an intrinsic term,foGo-B,and a"driving force"term, 0.5△E The model is general and has been described in detail From the semiempirical expression for the height of the crossing beforeand applied to a large number of reactions of dif. point,it was possible to derive the value of the f factor for a series ferent types.Here we will briefly summarize some VB computational of identity H abstraction reactions.As the ffactor appears to be applications on hydrogen abstraction reactions and various SN2 relatively constant and close to 1/3,eqs 42 and 49 can be turned reactions. into the very simple eq 50,where it is seen that the barrier for 3.2.1.Hydrogen Abstraction Reactions.Consider a gen- identity H abstraction reactions depends on a single parameter of eral hydrogen abstraction reaction that involves cleavage of a the reactants,D(X-H). bond H-Y by a radical X"(X,Y a univalent atom or a △E≈(2f-0.5)D(X-H)f=1/3 (50) molecular fragment): The so-calculated barriers were shown to correspond quite X+H-Y一X-H+Y (45) well to the corresponding CCSD(T)barriers for a series of dx.dol.org/10.1021/cr100228rChem.Rev.XXXX,XXX,000-000
L dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW This diagram applies to elementary reactions wherein the barrier can be described as the interplay of two major VB states, that of the reactants and that of the products. It displays the ground-state energy profile for the reacting system (bold curve), as well as the energy profiles for individual VB states (thinner curves); these latter curves are also called “diabatic”curves, while the full state energy curve (in bold) is called “adiabatic”. Thus, starting from the reactant geometry on the left, the VB structure Ψr that represents the reactant’s electronic state, R, has the lowest energy and merges with the ground state. Then, as one deforms the reacting molecules toward the product geometry, Ψr gradually rises and finally reaches an excited state P* that represents the VB structure of the reactants in the product geometry. A similar diabatic curve can be traced from P, the VB structure of the products in its optimal geometry, to R*, the same VB structure but in the reactant geometry. Consequently, the two curves cross somewhere in the middle of the diagram. The crossing is of course avoided in the adiabatic ground state, owing to the mixing of the two VB structures, which stabilizes the resulting transition state by a resonance energy term, labeled B. The barrier is thus interpreted as arising from avoided crossing between two diabatic curves which represent the energy profiles of the VB state curves of the reactants and products. The nature of the R* and P* promoted states depends on the reaction type and will be explained below using specific examples. In all cases, the promoted state R* is the electronic image of P in the geometry of R, while P* is the image of R at the geometry of P. The G terms are the corresponding promotion energy gaps, B is the resonance energy of the transition state, ΔEq is the energy barrier, and ΔErp is the reaction energy. The simplest expression for the barrier is given by ΔEq ¼ fGr B ð42Þ Here, the term fGr is the height of the crossing point, expressed as some fraction (f) of the promotion gap at the reactant side (Gr). A more explicit expression is ΔEq ≈ f0G0 þ ðGp=2G0ÞΔErp þ ð1=2G0ÞΔErp2 B G0 ¼ 0:5ðGr þ GpÞ f0 ¼ fr þ fp ð43Þ which considers the two promotion gaps and f factors through their average quantities, G0 and f0. Equation can be further simplified by neglecting the quadratic term and taking Gp/2G0 as ∼1/2, thus leading to ΔEq ¼ f0G0 þ 0:5ΔErp B ð44Þ Equation 44 expresses the barrier as a balance of the contributions of an intrinsic term, f0G0 B, and a “driving force” term, 0.5ΔErp. The model is general and has been described in detail before183187 and applied to a large number of reactions of different types. Here we will briefly summarize some VB computational applications on hydrogen abstraction reactions and various SN2 reactions. 3.2.1. Hydrogen Abstraction Reactions. Consider a general hydrogen abstraction reaction that involves cleavage of a bond HY by a radical X•v (X, Y = a univalent atom or a molecular fragment): X•v þ HY f XH þ Y•v ð45Þ Practically, Ψr is a linear combination of covalent and ionic forms that contribute to the Lewis structure “X•v + HY”, as shown in the following: Ψr ¼ C1ðX•v þ H• • YÞ þ C2ðX•v þ Hþ :YÞ þ C3ðX•v þ H: YþÞ ð46Þ This combination is maintained in Ψr from R to P* throughout the reaction coordinate, while the coefficients of the contributing structures change and adapt themselves to the geometric change (e.g., at infinite H---Y distance, C1 = 1). The curve Ψp, which stretches between P and R* is defined in an analogous manner. Two definitions are possible for the diabatic state curves Ψr and Ψp. In the variational diabatic configuration (VDC) method, the energies of the diabatic states are variationally minimized, as is done in the applications described below unless otherwise noted. In this way, each diabatic state has the best possible combination of coefficients and orbitals for this specific state. Alternatively, in the consistent diabatic configuration (CDC) method,188 the diabatic states are simply extracted from the ground-state wave function by projection. As explained before,183 the inconvenience of the CDC technique is that the so-constructed diabatic states involve orbitals and VB coefficients of the ground state and are therefore not optimal for the diabatic states. By comparison, the VDC technique gives quasi-variational quantities for all the parameters in the diagram (f, G, B). Since the promoted state R* is the VB structure of P in the geometry of R, its electronic state is illustrated by R ¼ ðX•v þ H• Þ---------• Y ð47Þ where the HY bond is infinitely long, while the X•v radical (spin-up) experiences some Pauli repulsion with the electron of H, which is 50% spin-up and 50% spin-down. As such, the R* state is 75% a triplet state, and hence, the Gr gap is (3/4)ΔEST, ΔEST being the singlettriplet excitation of the XH bond that undergoes activation. In the study of identity reactions (X = Y), it has been shown that, indeed, the promotion energy Gr required to go from R to R* is proportional to the singlettriplet gap of the XH bond186,189 or to the XH bonding energy D(XH). Actually, systematic VB ab initio calculations by the VBCI method have shown that, to a good approximation, Gr can be expressed as follows:189 Gr ≈ 2DðXHÞ ð48Þ Moreover, a semiempirical derivation showed189 that the resonance energy B is also proportional to D(XH) by the following expression: B ≈ 0:5DðXHÞ ð49Þ From the semiempirical expression for the height of the crossing point, it was possible to derive the value of the f factor for a series of identity H abstraction reactions. As the f factor appears to be relatively constant and close to 1/3, eqs 42 and 49 can be turned into the very simple eq 50, where it is seen that the barrier for identity H abstraction reactions depends on a single parameter of the reactants, D(XH). ΔEq ≈ ð2f 0:5ÞDðXHÞ f ¼ 1=3 ð50Þ The so-calculated barriers were shown to correspond quite well to the corresponding CCSD(T) barriers for a series of
Chemical Reviews REVIEW △F(eq44) △E(eq52) 40 451 35 (a) (b) 35 30 R2=0.9900 喝 R2=0.9734 25 20 20 15- 15 10 5 1015202530354045 51015202530354045 △E(ab initio VB, △E(ab initio VB】 Figure 6.VBSCD-derived barriers plotted against ab initio VB-calculated barriers (kcal/mol). identity abstraction reactions (X=Y=H,CH3,SiH3,GeH At each point of the reaction coordinate,the three diabatic SnH3,PbH3).s9 While the limitations of this expression have VB structures,respectively those of the reactants and products, been discussed in detail,e.g.,in the case where the transition state complemented by an ionic structure [H3N:H':NH3],were nogoicr袖3h2qogoa8ioR calculated by the BLW method,and the adiabatic ground state mixing ofadditional structures,1 was calculated by configuration interaction between these VB magnitude and correctly reproduces the trends in the series. structures,without further optimization of the orbitals.The For nonidentity reactions (XY),eq 44,which accounts MOVB results were found to be in good accord with the explicitly for all the reactivity factors,was applied to estimate corresponding ab initio Hartree-Fock calculations for the pro- the barriers for a series of 14 H abstraction reactions.The plot ton transfer process.The authors also incorporated solvent in Figure 6a shows good agreement with VBCISD-computed effects into the MOVB Hamiltonian like in QM/MM calculations barriers. and have modeled the proton transfer between ammonium ion Albeit being apparently more difficult to treat than identity and ammonia in water using statistical Monte Carlo simulations reactions,if one assumes that the transition state coincides with in a cubic cell involving 510 water molecules with periodic the geometry of the lowest crossing point in the VBSCD,one can boundary conditions.25 By comparison to previous semiempi- treat nonidentity reactions using semiempirical approximations rical treatments,both diagonal and off-diagonal matrix that enable estimation of barriers from easily available quantities. elements in the MOVB Hamiltonian explicitly include solvent Since the A-X and A-Y bonds are of different strengths,one effects in the calculation.The reaction coordinate in solution is being the weakest and the other the strongest,they have chosen as the energy difference between the diabatic reactant and respective bonding energies Dw and Ds.VB calculations for the product VB states,which ensures that the solvent degrees of 14 reactions showed that B is approximately half of the weakest freedom are adequately defined because the change in solute bonding energy,Dw,or,in other words,of the bond energy of -solvent interaction energy reflects the collective motions of the the bond that is broken in the reactants of the exothermic solvent molecules as the reaction.2 As a result,solvent effects direction of the reaction,while Go is close to the sum of both were found to increase the barrier by 2.2 kcal/mol relative to that bonding energies:19 of the gas-phase process. B=0.5Dw Go =Dw Ds (51) 3.2.2.SN2 Reactions in the Gas Phase.A generic SN2 reaction is shown in eq 54 where the nucleophile,X:,shifts an Thus,by taking fo1/3 as in identity reactions(accurate VB electron to the A-Y electrophile and forms a new X-A bond calculations yield fo0.32-.6) one gets the following while the leaving group Y departs with the negative charge. very simple equation: X+A-Y→X-A+Y (54) △E=K(Ds-0.5Dw)+0.5△Ep K≈1/3 (52) Let us derive now an expression for G,by simply examining the Figure 6b displays a good correlation of the barriers calculated through eq 52 for the same 14 reactions as in Figure 6a plotted omeca against the VBCISD calculations.Thus,it appears that the each other(as in the ground state,R)and separated from X by a VBSCD model is able to express semiquantitative barriers for long distance.The X fragment,which is neutral in the product P, H abstraction reactions in terms of the bonding energies of must remain neutral in R*and therefore carries a single active reactants and products.Recent applications of eq 52 to the electron.As a consequence,the negative charge is located on the reactivity of cytochrome P450 in alkane hydroxylation shows that A---Y complex,so that the R*state is the result ofa charge transfer a good correlation with DFT-computed barriers is achieved with from the nucleophile (X:)to the electrophile (A-Y),as depicted by 0.3 and a constant B value,which is very close to R*=X*//(A..Y) (55) The MOVB method(see section 2.4.2)has also been applied by Mo and Gao to describe the diabatic and adiabatic potential It follows that the promotion from R to R*has two parts:an energy curves in a model proton abstraction reaction: electron detachment from the nucleophile,X:,and an electron attachment to the electrophile,A-Y.The promotion energy G HN:+H-NH+一HNt-H+NH3 (53) is therefore the difference between the vertical ionization dx.dol.org/10.1021/cr100228r Chem.Rev.XXXX,XXX,000-000
M dx.doi.org/10.1021/cr100228r |Chem. Rev. XXXX, XXX, 000–000 Chemical Reviews REVIEW identity abstraction reactions (X = Y = H, CH3, SiH3, GeH3, SnH3, PbH3).189 While the limitations of this expression have been discussed in detail, e.g., in the case where the transition state is not colinear for which B is larger than its value in eq 49 due to mixing of additional structures,183 still eq 50 yields good orders of magnitude and correctly reproduces the trends in the series. For nonidentity reactions (X 6¼ Y), eq 44, which accounts explicitly for all the reactivity factors, was applied190 to estimate the barriers for a series of 14 H abstraction reactions. The plot in Figure 6a shows good agreement with VBCISD-computed barriers. Albeit being apparently more difficult to treat than identity reactions, if one assumes that the transition state coincides with the geometry of the lowest crossing point in the VBSCD, one can treat nonidentity reactions using semiempirical approximations that enable estimation of barriers from easily available quantities. Since the AX and AY bonds are of different strengths, one being the weakest and the other the strongest, they have respective bonding energies DW and DS. VB calculations for the 14 reactions showed that B is approximately half of the weakest bonding energy, DW, or, in other words, of the bond energy of the bond that is broken in the reactants of the exothermic direction of the reaction, while G0 is close to the sum of both bonding energies:190 B ¼ 0:5DW G0 ¼ DW þ DS ð51Þ Thus, by taking f0 ≈ 1/3 as in identity reactions (accurate VB calculations yield f0 = 0.320.36),190 one gets the following very simple equation: ΔEq ¼ KðDS 0:5DWÞ þ 0:5ΔErp K ≈ 1=3 ð52Þ Figure 6b displays a good correlation of the barriers calculated through eq 52 for the same 14 reactions as in Figure 6a plotted against the VBCISD calculations. Thus, it appears that the VBSCD model is able to express semiquantitative barriers for H abstraction reactions in terms of the bonding energies of reactants and products. Recent applications of eq 52 to the reactivity of cytochrome P450 in alkane hydroxylation shows that a good correlation with DFT-computed barriers is achieved with eq 52 using f = 0.3 and a constant B value, which is very close to 0.5DW. 191,192 The MOVB method (see section 2.4.2) has also been applied by Mo and Gao to describe the diabatic and adiabatic potential energy curves in a model proton abstraction reaction:125 H3N: þ HNH3 þ f H3NþH þ :NH3 ð53Þ At each point of the reaction coordinate, the three diabatic VB structures, respectively those of the reactants and products, complemented by an ionic structure [H3N: H+ :NH3], were calculated by the BLW method, and the adiabatic ground state was calculated by configuration interaction between these VB structures, without further optimization of the orbitals. The MOVB results were found to be in good accord with the corresponding ab initio HartreeFock calculations for the proton transfer process. The authors also incorporated solvent effects into the MOVB Hamiltonian like in QM/MM calculations and have modeled the proton transfer between ammonium ion and ammonia in water using statistical Monte Carlo simulations in a cubic cell involving 510 water molecules with periodic boundary conditions.125 By comparison to previous semiempirical treatments,174,175 both diagonal and off-diagonal matrix elements in the MOVB Hamiltonian explicitly include solvent effects in the calculation. The reaction coordinate in solution is chosen as the energy difference between the diabatic reactant and product VB states, which ensures that the solvent degrees of freedom are adequately defined because the change in solute solvent interaction energy reflects the collective motions of the solvent molecules as the reaction.125 As a result, solvent effects were found to increase the barrier by 2.2 kcal/mol relative to that of the gas-phase process. 3.2.2. SN2 Reactions in the Gas Phase. A generic SN2 reaction is shown in eq 54 where the nucleophile, X:, shifts an electron to the AY electrophile and forms a new XA bond while the leaving group Y departs with the negative charge. X: þ AY f XA þ :Y ð54Þ Let us derive now an expression forGr by simply examining the nature of the excited state R* relative to the corresponding ground state.183187 In R*, A and Y are geometrically close to each other (as in the ground state, R) and separated from X by a long distance. The X fragment, which is neutral in the product P, must remain neutral in R* and therefore carries a single active electron. As a consequence, the negative charge is located on the A---Y complex, so that the R* state is the result of a charge transfer from the nucleophile (X:) to the electrophile (AY), as depicted by R ¼ X• // ðA\YÞ ð55Þ It follows that the promotion from R to R* has two parts: an electron detachment from the nucleophile, X:, and an electron attachment to the electrophile, AY. The promotion energy Gr is therefore the difference between the vertical ionization Figure 6. VBSCD-derived barriers plotted against ab initio VB-calculated barriers (kcal/mol)