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J. Phys. Chem. A 2010, 114, 1980–1984
Nonadiabatic Histidine Dissociation of Hexacoordinate Heme in Neuroglobin Protein Feng Zhang,†,‡ Yue-Jie Ai,†,‡ Yi Luo,*,† and Wei-Hai Fang*,‡ Theoretical Chemistry, School of Biotechnology, Royal Institute of Technology, S-10691 Stockholm, Sweden, and College of Chemistry, Beijing Normal UniVersity, Beijing 100875, China ReceiVed: October 15, 2009; ReVised Manuscript ReceiVed: December 2, 2009
In the present work, density functional theory and canonical nonadiabatic Monte Carlo transition state theory have been used to investigate the histidine dissociation process from hexacoordinate heme in Ngb protein. The potential energy surfaces (PES) of the lowest singlet, triplet, and quintet states are calculated by stepwise optimization along with the histidine dissociation pathway. Based on the calculated two-dimensional PES, the histidine dissociation rates for the spin-forbidden processes via singlet to triplet and singlet to quintet transitions have been calculated by the nonadiabatic Monte Carlo transition state theory in canonical ensemble. The present study provides a quantitative description on spin-forbidden histidine dissociation processes. Introduction Heme-containing globin proteins continue to be one of the most-studied classes of biomolecules due to their diverse functions as sensors, activators, and transporters of gaseous molecules. The most widely studied examples are hemoglobin (Hgb) and myoglobin (Mgb), which contain a pentacoordinated heme iron. The pentacoordinated heme iron provides an open ligand binding site, which has been regarded as the common characteristic for heme proteins.1-4 Nevertheless, a new class of heme protein with an unusual hexacoordinated heme iron was discovered one decade ago,5-11 breaking people’s prevenient view. The hexacoordinate hemoglobins exhibit more complex binding kinetics than pentacoordinate hemoglobins, due to the reversible intramolecular coordination by a histidine side chain.12 Neuroglobin (Ngb) is a representing substance in the family of hexacoordinated hemoglobins. The dissociation of distal histidine (His64) was experimentally suggested to be the first step, which is slower than the subsequent ligand binding processes.6,8,13-18 Distinct values of histidine dissociation rates (8200, 4.5, and 0.6 s-1) with a major difference of a factor of 103 were inferred by different experimental studies, although the experimental conditions were similar.6,8,18 As a result, the internal His64 affinity (ratio of His64 association and dissociation rate) was deduced to be 1.2, 444.4, and 3000 for human Ngb in the previous studies. The ground state of the hexacoordinated heme is a singlet state (S0), whereas the triplet and quintet states become lower than the singlet state in energy for the pentacoordinated intermediate. Thus, the dissociation of His64 is a nonadiabatic process, which has an important influence on the rate of the His64 dissociation. Numerous experimental studies have been carried out to explore the kinetic behavior of the His64 dissociation and subsequent binding processes of Ngb.5-22 The density functional theory (DFT) and some high-level ab initio methods have been extensively used to study the electronic structures and kinetics of pentacoordinate hemeproteins and their model systems, most of which focused on the pentacoordinate Hgb or Mgb.23-31 For * Corresponding authors: Email:
[email protected] (Y.L),
[email protected]. (W.-H.F.). Telephone: +46-8-55378414 (Y.L.), +86-10-58805382 (W.-H.F). Fax: +46-8-55378590 (Y.L.). † Royal Institute of Technology. ‡ Beijing Normal University.
example, Harvey et al.25-28 have investigated the small ligand (O2, CO, NO, and H2O) binding to pentacoordinated models of heme with the DFT, CASSCF, CASPT2, and CCSD(T) methods, which mainly focused on the minimum-energy crossing points and their roles in the nonadiabatic binding and dissociation processes. It is surprising that only few theoretical studies concern the mechanical and kinetics aspects of the hexacoordinated hemoglobins. Recently, the DFT and hybrid DFT/MM methods were used to investigate CO binding to the hexacoordinate heme in Ngb protein.32,33 Two nonadiabatic sequential pathways were determined to be important for CO binding to the hexacoordinate heme, which involve the singlet/triplet and singlet/quintet intersections.32,33 Although ab initio calculations provided a qualitative description on the histidine dissociation, the rate of the His64 dissociation has not been calculated in these previous studies. The disagreement in the experimentally inferred rate is not investigated from viewpoint of theory up to date. In this work, the canonical nonadiabatic Monte Carlo transition state theory (NA-MCTST) was used to simulate the kinetic behavior of the His64 dissociation, which is based on the DFT calculated oneand two-dimensional potential energy surfaces. Computational Methods Ab Initio Calculations. DFT with B3LYP exchange correction functional has previously been used to explore structure, binding, and dissociation characters of the pentacoordinated heme systems, and the calculated results reproduced experimental structures and energies with good accuracy.23-31,34-37 All quantum chemical calculations were performed with the DFT (B3LYP) method in this work (The unrestricted DFT method was used for open-shell systems). The effective core potential with double-ξ valence basis (LanL2DZ)38 was used to describe the iron atom. The 6-311G** basis set was applied for all other atoms in the molecule. The mixed basis sets of LanL2DZ and 6-311G** is referred to as GEN hereafter. All the calculations were performed with Gaussian 03 package of programs.39 In this work, we focused mainly on the kinetics of the His64 dissociation process in Ngb. Therefore, a molecule composed of a porphyrin ring with a hexa-coordinated iron at the center and two imidazole rings ligated to the iron was chosen as the model system. Despite the natural reaction coordinate-the
10.1021/jp909887d 2010 American Chemical Society Published on Web 01/05/2010
Histidine Dissociation of Hexaccordinate Heme
J. Phys. Chem. A, Vol. 114, No. 4, 2010 1981
distance between His64 and the heme iron (RFe-N), the out-ofplane movement of iron atom relative to the porphyrin ring (RFe-P) was considered as another substantial factor involved in the ligand binding process of heme protein.24,32 The adiabatic potential curves (PEC) of the His64 dissociation on three different electronic states were stepwise optimized at the B3LYP/GEN level with the RFe-N distance fixed at different values. Meanwhile, two-dimensional (2D) potential energy surfaces for the three electronic states (PESs) were scanned as a function of RFe-N and RFe-P distances with all other bond parameters fully optimized. The obtained 1D and 2D PES were used to calculate the rate constant of the spin-forbidden His64 dissociation in the present study. Monte Carlo Transition State Theory. Different multidimensional models were employed to study intersystem crossing processes.40,41 Zhao and Nakamura42 have developed the NAMCTST method for calculation of the thermal rate constant of complicated molecular systems, which is improved for the present system. The advantage of this method was that the crossing seam surface was taken into account to contribute to the nonadiabatic rate instead of considering only one crossing point. In addition, the Monte Carlo summation was expressed in configuration space not in phase space, which provided a convenient way to carry out Monte Carlo simulation. The standard canonical adiabatic reaction rate constant is described as42,43
kZr ) lim tf∞
1 h3N
∫ dP dQ eβHPT∇S(Q)δ[S(Q)]h{S[Q(t)]} (1)
where k is the adiabatic rate constant, Zr is the partition function of the reactant, H is the Hamiltonian of the system, h(x) is Heaviside function, and S(Q) defines the dividing surface. To consider the nonadiabatic transition, one can substitute h{S[Q(t)]} in eq 1 by the nonadiabatic transition probability P[E⊥,S(Q)], which could be calculated with the Landau-Zener formula44,45 or other expressions.46-48 To calculate the intersystem crossing (ISC) rate of the histidine dissociation, the Landau-Zener probability was used in this work.
PLZ )
2πV212 h∆F
µ⊥ 2E⊥
2π β
1 h3N (3N-1)/2
( )
1 h3N
∫ dP dQ e-βH ) h13N ( 2π ∫ dQ e-βV(Q) β) 3N/2
(4) Combine eq 4 with eq 3 and we get
k)
∫ dQ e-βV(Q)∇S(Q)δ[S(Q)] ∫ e-βE PLZ[E⊥, S(Q)]dE⊥ ∫ dQ e-βV(Q) ⊥
β 2π
(5) If the distribution function f is defined as
f)
∫ dQ e-βV(Q)∇S(Q) ∫ e-βE PLZ[E⊥, S(Q)]dE⊥ ⊥
(3) Similarly, the partition function in canonical ensemble can be written as
e-βV(Q)
(6)
∫ dQ e-βV(Q)
it comes the Monte Carlo summation of eq 3:
k)
2πβ |2ε|1 N1 ∑ δ ∇S(Q ) ∫ e i
i
i
∞ -βE⊥
0
PLZ[E⊥, S(Q)]dE⊥
(7)
here β ) 1/kBT, where kB is the Boltzmann constant. 2ε is the chosen energy “shell” around S(Q) (used to approximate the delta function (δ[S(Q)]). The statistical weight function for Monte Carlo simulation is chosen as
W(Qi) ) e-βV(Qi)
(8)
according to the distribution function in eq 6. However, acceptance of low energy configuration is favored with the above statistical weight and sample configuration lying in the seam region becomes rare event. To overcome this difficulty, the importance sampling method50,51 should be used in this case. The importance weight is chosen as
(2)
where µ⊥ is the reduced mass along the direction orthogonal to the crossing seam, V12 is the spin-orbit coupling for the ISC process; h is the Planck constant, and (E-EC) is the available kinetic energy in the coordinate normal to the seam. ∆F is the norm of the difference of the gradients, |∂V1/∂q - ∂V2/∂q|. The integral in eq 1 can be expressed in the configuration space by separating variables,36,49 then the canonical nonadiabatic rate becomes,
kZr )
Zr )
P0(Qi) ) eβV(Qi)
(9)
As a result, the Markov walk is performed with each move being weighted equally. To calculate the final MC sum, each selected configuration is then weighted by 1/P0. The appropriate MC sum eq 8 with importance sampling becomes
k(T) )
{
2πβ |2ε|1 N1
∑ δi∇S(Qi) ∫0
∞ -βE⊥
e
P[E⊥, S(Qi)]dE⊥[P-1 0 (Qi)]
i
-1
N
N
∑ i)1
[P-1 0 (Qi)]
}
(10)
Based on the 1D and 2D PES constructed by the DFT/GEN calculations, the nonadiabatic rate constant in eq 10 was obtained by the following Monte Carlo simulation procedure. Starting with the model molecule in its ground state equilibrium
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Zhang et al.
Figure 1. Schemetic structures for the S0, T1, and Q1 states along with the relative energy in parentheses
geometry, a total of Ninc ) 1 × 105 random moves are first made as an incubation period to eliminate the influence of the initial configuration, and subsequently N ) 1 × 106 states are generated to compute the MC sum in eq 10. Our ab initio PES provide the optimal pathway, therefore only the two imidazole groups and Fe atom move along the Nd-Fe-Np axe during the Markova move, while the other bond parameters are deemed to be at optimum values. The statistical weight for each random configuration is equal, resulting from the importance sampling method. The reactant configuration space for the histidine dissociation of the model system is defined as those configurations having 2.0 Å e RFe-N e 3.9 Å, 0.0 Å e RFe-P e 0.4 Å, corresponding to ordinary bond dissociation reactions. Here RFe-N is the distance of distal histidine to the Fe atom, and RFe-P is the movement of the Fe atom out of the porphyrin plane. For the configuration falling inside of the reactant configuration space, the energies of S0, T1, and Q1 electronic states of the configuration are checked to determine if it is close to the seam. If the energy difference |V1 - V2| is less than a specified energy width 2ε, the configuration is deemed to lie in the crossing seam. Here, ε is defined in accordance with the spin-orbit coupling. The spin-orbit coupling value for heme-CO model was estimated to be 10 cm-1 for S/Q interaction27,52,53 and 2.5 cm-1 for S/T53 interaction. It is reasonable to expect that similar spin-orbit coupling constant exists for the present Ngb model, because the part of ferrous heme plays a completely determining role for the ISC process. For each crossing point selected by the program, the spin-orbit coupling is assumed to be constant, which is acceptable because the electronic nature of the two potential energy surfaces is not changing much in the narrow region of the crossing space.35 Results and Discussions Potential Energy Surfaces. The hexacoordinated heme in the lowest singlet (S0), triplet (T1), and quintet (Q1) states were optimized at the B3LYP/GEN level, and the obtained structures are shown in Figure 1 along with the three important bond parameters involved in the distal imidazole dissociation. The relative energies of three equilibrium structures are also given in Figure 1. The T1 and Q1 states are nearly degenerate with energy difference less than 0.5 kcal/mol. Besides the distance between the distal imidazole ring and Fe atom, RFe-N, the movement of Fe out of the porphyrin plane, referred to as RFe-P, and the distortion angel of two imidazole rings, referred to as AIm-Im hereafter, could influence the dissociation process. It can be seen that the two imidazole rings are nearly perpendicular to each other in the S0 structure, whereas they are almost parallel in the T1 and Q1 states. The single point energies were calculated at the optimized singlet and triplet structures with AIm-Im at 180° and 90°, respectively. It was found the energy difference is very
Figure 2. Potential energy curves for the histidine dissociation by B3LYP/GEN stepwise optimization
small and the orientation of the two imidazole rings has little effect on the potential energy of the system. Therefore, the orientation changes of the two imidazole rings are excluded from the construction of the potential energy surface in the present study. The significant secondary coordinate is the movement of Fe atom out of the porphyrin plane. The Fe atom is located inside the porphyrin plane at the optimized equilibrium structures of the S0, T1, and Q1 states. However, the movement of Fe in the His64 dissociation presents distinct characteristic for the three electronic states. From the equilibrium configuration to the pentacoordinated heme where Fe-N distance was assumed to be 3.9 Å in this work, the Fe atom departs from the porphyrin plane by 0.12, 0.16, and 0.36 Å for the S0, T1, and Q1 states, respectively. The 2D PES were scanned as a function of the RFe-N and RFe-P distances with the other bond parameters optimized fully. The RFe-P distance is changed in the range of 0.0-0.18 Å for the S0 and T1 states and in the range of 0.0-0.37 Å for the S0 and Q1 states with the step size of 0.035 Å. Meanwhile, the RFe-N distance is changed from 2.0 Å to 3.9 Å with the step size of 0.1 Å for the three electronic states. The numerical method of 2D cubic spline interpolation was applied to obtain smooth figures in Figure 3 (panels a and b). It is worthy to mention that the crossing space of S0/T1 and S0/Q1 were calculated with a small step-size (0.01 Å) for 1D and 2D cases, as shown in the insets of Figure 2 for the 1D case. When the movement Fe atom out of the porphyrin ring is considered, the S0/T1 crossing point is located at RFe-N ) 2.64 Å and RFe-P ) 0.07 Å, and the S0/Q1 crossing point is at RFe-N ) 2.72 Å and RFe-P ) 0.11 Å. In the His64 dissociation process, the Fe atom moves out of the porphyrin plane more in the Q1 state than in the S0 and T1 states. However, the configurations with RFe-P > 0.2 Å contribute little to the S0 f Q1 nonadiabatic transition.
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J. Phys. Chem. A, Vol. 114, No. 4, 2010 1983
Figure 3. Two-dimensional potential energy surfaces of the S0, T1, and Q1 states with respect to the distances of Fe atom to histidine and to porphyrin plane. (a) S0 and T1 states; (b) S0 and Q1 states.
Figure 4. Arrhenius plot of log k (in s-1) versus inverse T, computed for nonadiabatic transition along histidine dissociation pathway. ST(Q)Adia-1D: the SfT(Q) transition rate calculated on the basis of 1D PES by the eq 10 with PLZ equal to unity; ST(Q)-Dia-1D: the SfT(Q) transition rate calculated on the basis of 1D PES by the eq 10; ST(Q)Dia-2D: the SfT(Q) transition rate calculated on the basis of the 2D PES by eq 10.
Monte Carlo Simulation. The Monte Carlo simulations, as mentioned above, were performed based on the 2D PES by the B3LYP/GEN calculations. For comparison, we calculated the adiabatic rate for the 1D case by assuming PLZ in eq 10 is equal to unity. The calculated rate constants of nonadiabatic S0fT1 and S0fQ1 transitions are shown in Figure 4 along with the adiabatic rates. The Arrhenius plots for the adiabatic and nonadiabatic rates are almost linear, and the calculated nonadiabatic rate curves for S0/T1 and S0/Q1 are nearly parallel to each other. The ratio of computed adiabatic and nonadiabatic rate is of 104, which means the average magnitude of Landau-Zener probability is of 10-4. As can be seen from the nonadiabatic rates, the computed S0fT1 and S0fQ1 transition rates are increased by around 2.5 and 5.0 times from 1D to 2D PES, respectively. There is another important information that can be read from the comparison between the calculated S0fT1 and S0fQ1 transition rates. The S0fQ1 transition occurs more quickly than the S0fT1 transition due to stronger spin-orbit coupling between the S0 and Q1 states. The quintet pentacoordinated heme is prior to the triplet one as a product of histidine dissociation. At room temperature (T ) 300 K), the rate constant of the S0fQ1 transition is calculated to be 1.34 × 104 s-1 by the nonadiabatic Monte Carlo simulation on the basis of the 2D PES, while the S0fT1 transition rate is predicted to be 1.99 × 103 s-1 at the same condition. The present calculations agree
with the experimentally inferred His64 dissociation rate for human Ngb by Trent et al.6 in the order of magnitude, which claimed that the histidine dissociation occurs on a time scale of milliseconds. Solvent and the protein environment could have considerable influence on the nonadiabatic His64 dissociation. A hybrid DFT/ MM method has been used to investigate CO binding to the hexacoordinate heme in the Ngb protein environment.32 The DFT/MM optimized bond parameters for the hexacoordinated and pentacoordinated hemes in the Ngb protein environment are close to those reported in the present work. A comparison reveals that the relative energies of stationary structures in the triplet and quintet states are reduced by ∼1.0 kcal/mol due to the protein environment. These give us a reason to expect that the protein environment has only a minor influence on the calculated rate constants of the nonadiabatic His64 dissociation for the present model system. The dissociation step of COligated heme has been characterized by means of DFT and TDDFT calculations.35,36 It was found that use of small model complexes can reproduce experimental data obtained with the complete Mb protein.36 Summary In the present work, density functional theory and canonical nonadiabatic Monte Carlo transition state theory have been used to investigate the histidine dissociation processes from the hexacoordinate heme in Ngb protein. The PES for the lowest singlet, triplet, and quintet states along the histidine dissociation pathway were obtained by stepwise optimization. The 2D PES were constructed by single-point energy calculations at each grid with considering the effect of the movement of the Fe atom out of the porphyrin plane as the second coordinate. The T1 and Q1 states are found to be degenerated in a wide range. In the histidine dissociation process, Fe atom moving out of the porphyrin plane in the Q1 state has a larger degree than in the S0 and T1 states. Based on the ab initio PES, the rate constants of spin-forbidden histidine dissociation via S0fT1 and S0fQ1 transitions were studied by the nonadiabatic Monte Carlo transition state theory in canonical ensemble. Compare with the 1D calculation, the effect of 2D PES increased the Monte Carlo rate by several times for the nonadiabatic transition. Our model study indicates that the quintet pentacoordinate heme is prior to the triplet as the product of histidine dissociation. At room temperature, the rate constant of S0fQ1 transition is 1.34 × 104 s-1 by our 2D nonadiabatic Monte Carlo simulation, while the S0fT1 transition rate is 1.99 × 103 s-1. Our calculations
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agree with the experimental histidine dissociation rate for human Ngb by Trent et.al.6 in the order of magnitude, and also agree with the DFT study recently performed by our group.32 The Monte Carlo simulations based the 2D ab inito PES in this work provides a quantitative description on the spin-forbidden histidine dissociation processes. Acknowledgment. This work was supported by grants from the NSFC (Grant 20720102038) and from the Major State Basic Research Development Programs (Grant 2004CB719903). References and Notes (1) Perutz, M. F. Nature 1970, 228, 726–739. (2) Austin, R.; Beeson, K.; Eisenstein, L.; Frauenfelder, H.; Gunsalus, C. Biochemistry 1975, 14, 5355–5373. (3) Olson, J. S.; Phillips, G. N. J. Biol. Chem. 1996, 271, 17593–17602. (4) Springer, B. A.; Olson, J. S.; Phillips, G. N. Chem. ReV. 1994, 94, 699–714. (5) Burmester, T.; Weich, B.; Reinhardt, S.; Hankeln, T. Nature 2000, 407, 520–523. (6) Trent III, J. T.; Watt, R. A.; Hargrove, M. S. J. Biol. Chem. 2001, 276, 30106–30110. (7) Couture, M.; Burmester, T.; Hanker, T.; Rousseau, D. J. Biol. Chem. 2001, 276, 36377–36382. (8) Dewilde, S.; Kiger, L.; Burmester, T.; Hanken, T.; Baudin-Creuza, V.; Aerts, T.; Marden, M. C.; Caubergs, R.; Moens, L. J. Biol. Chem. 2001, 276, 38949–38955. (9) Trent III, J. T.; Hargove, M. S. J. Biol. Chem. 2002, 277, 19538– 19545. (10) Burmester, T.; Ebner, B.; Weich, B.; Hankeln, T. Mol. Biol. E Vol. 2002, 19, 416–421. (11) Pesce, A.; Bolognesi, M.; Bocedi, A.; Ascenzi, P.; Dewilde, S.; Moens, L.; Hankeln, T.; Burmester, T. EMBO Rep. 2002, 3, 1146–1151. (12) Hargrove, M. S. Biophys. J. 2000, 79, 2733–2738. (13) Trent III, J. T.; Hvitved, A. N.; Hargrove, M. S. Biochem 2001, 40, 6155–6163. (14) Hvitved, A. N.; Trent III, J. T.; Premer, S. A.; Hargrove, M. S. J. Biol. Chem. 2000, 276, 34714–34721. (15) Kriegl, J. M.; Bhattacharyya, A. J.; Nienhaus, K.; Deng, P.; Minkow, O.; Nienhaus, G. U. Proc. Natl. Acad. Sci. 2002, 99, 7992–799. (16) Sanctis, D.; Pesce, A.; Nardini, M.; Bolognesi, M.; Bocedi, A.; Ascenzi, P. IUBMB Life 2004, 56, 643–651. (17) Uno, T.; Ryu, D.; Tsutsumi, H.; Tomisugi, Y.; Ishikawa, Y.; Wilkinson, A. J.; Sato, H.; Hayashi, T. J. Biol. Chem. 2004, 279, 5886– 58935. (18) Kiger, L.; Uzan, J.; Dewilde, S.; Burmester, T.; Hankeln, T.; Moens, L.; Hamdane, D.; Baudin-Creuza, V.; Marden, M. C. IUBMB life 2004, 6, 709–719. (19) Nienhaus, K.; Kriegl, J. M.; Nienhaus, G. U. J. Biol. Chem. 2004, 279, 22944–22952. (20) Couture, M.; Burmester, T.; Hankeln, T.; Rousseau, D. L. J. Biol. Chem. 2001, 276, 36377–36382. (21) Van Doorslaer, S.; Dewilde, S.; Kiger, L.; Nistor, S. V.; Goovaerts, E.; Marden, M. C.; Moens, L. J. Biol. Chem. 2003, 278, 4919–4925.
Zhang et al. (22) Pesce, A.; De Sanctis, D.; Nardini, M.; Dewilde, S.; Moens, L.; Hankeln, T.; Burmester, T.; Ascenzi, P.; Bolognesi, M. IUBMB Life 2004, 56, 657–664. (23) McMahon, B. H.; Stojkovic, B. P.; Hay, P. J.; Martin, R. L.; Garcia, A. E. J. Chem. Phys. 2000, 113, 6831–6850. (24) Franzen, S. Proc. Natl. Acad. Sci. 2002, 99, 16754–16759. (25) Harvey, J. N. J. Am. Chem. Soc. 2000, 122, 12401–12402. (26) Harvey, J. N.; Poli, R.; Smith, K. M. Coord. Chem. ReV. 2003, 238, 347–361. (27) Harvey, J. N. Faraday Discuss. 2004, 127, 165–177. (28) Strickland, N.; Harvey, J. N. J. Phys. Chem. B. 2007, 111, 841– 852. (29) Jensen, K. P.; Ryde, U. J. Biol. Chem. 2004, 279, 14561–14569. (30) Radon, M.; Pierloot, K. J. Phys. Chem. A. 2008, 112, 11824–11832. (31) Chen, H.; Ikeda-Saito, M.; Shaik, S. J. Am. Chem. Soc. 2008, 130, 14778–14790. (32) Juan, L.; Ai, Y.-J.; Xie, Z.-Z.; Fang, W.-H. J. Phys. Chem. B. 2008, 112, 8715–8723. (33) Ming, X.; Fang, W.-H. J. Phys. Chem. B. 2008, 112, 990–996. (34) Freindorf, M.; Shao, Y.; Brown, S. T.; Kong, J.; Furlani, T. R. Chem. Phys. Lett. 2006, 419, 563–566. (35) Dreuw, A.; Dunietz, B. D.; Head-Gordon, M. J. Am. Chem. Soc. 2002, 124, 12070–12071. (36) Dunietz, B. D.; Dreuw, A.; Head-Gordon, M. J. Phys. Chem. B. 2003, 107, 5623–5629. (37) Xu, C.-L.; Ibrahim, M.; Spiro, T. G. Biochemistry 2008, 47, 2379– 2387. (38) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 299–310. (39) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M.; Cheeseman, A. J. R.; Zakrzewski, V. G.; Montgomery, J. A.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M; Cammi, R.; Mennucci, B.; Promelli, C., Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Cioslowski, J.; Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; Martin, R. L., Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M. W.; Johnson, B. G.; Chen, W.; Wong, M. W.; Andres, J. L.; Head-Gordon, M., Replogle, E. S. Pople, J. A. Gaussian 03, ReVision D.01; Gaussian, Inc.: Wallingford CT, 2004. (40) Truhlar, D. G. Faraday Discuss. 1998, 110, 521–535. (41) Cui, Q.; Morokuma, K.; Bowman, J. M.; Klippenstein, S. J. J. Chem. Phys. 1999, 110, 9469–9482. (42) Zhao, Y.; Mil’nikov, G.; Nakamura, H. J. Chem. Phys. 2004, 121, 8854–8860. (43) Miller, W. H.; Schwartz, S. D.; Tromp, J. W. J. Chem. Phys. 1983, 79, 4889–4898. (44) Landau, L. D. Phys. Z. Sowjetunion 1932, 2, 46. (45) Zener, C. Proc. R. Soc. London, Ser. A. 1932, 137, 696. (46) Delos, J. B. J. Chem. Phys. 1973, 59, 2365–2369. (47) Zhu, C.; Nakamura, H. J. Chem. Phys. 1994, 101, 10630–10647. (48) Zhu, C.; Nakamura, H. J. Chem. Phys. 1995, 102, 7448–7461. (49) Voter, A. F.; Doll, J. D. J. Chem. Phys. 1984, 80, 5832–5838. (50) Marks, A. J.; Thompson, D. L. J. Chem. Phys. 1992, 96, 1911– 1918. (51) Hastings, W. K. Biometrika. 1970, 57, 97–109. (52) Frauenfelder, H.; Wolynes, P. G. Science 1985, 229, 337. (53) Redi, M. H.; Gerstman, B. S. Biophys. J. 1981, 35, 471–484.
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