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Review
, 117 (3), 1564-1686

Metal Ion Modeling Using Classical Mechanics

Affiliations
Review

Metal Ion Modeling Using Classical Mechanics

Pengfei Li et al. Chem Rev.

Abstract

Metal ions play significant roles in numerous fields including chemistry, geochemistry, biochemistry, and materials science. With computational tools increasingly becoming important in chemical research, methods have emerged to effectively face the challenge of modeling metal ions in the gas, aqueous, and solid phases. Herein, we review both quantum and classical modeling strategies for metal ion-containing systems that have been developed over the past few decades. This Review focuses on classical metal ion modeling based on unpolarized models (including the nonbonded, bonded, cationic dummy atom, and combined models), polarizable models (e.g., the fluctuating charge, Drude oscillator, and the induced dipole models), the angular overlap model, and valence bond-based models. Quantum mechanical studies of metal ion-containing systems at the semiempirical, ab initio, and density functional levels of theory are reviewed as well with a particular focus on how these methods inform classical modeling efforts. Finally, conclusions and future prospects and directions are offered that will further enhance the classical modeling of metal ion-containing systems.

Conflict of interest statement

The authors declare no competing financial interest.

Figures

Figure 1
Figure 1
Active molecular orbitals forming the chemical bond between two uranium atoms. The orbital label is given below each orbital, together with the number of electrons occupying this orbital or pair of orbitals in the case of degeneracy. Reprinted with permission from ref (39). Copyright 2005 Nature Publishing Group.
Figure 2
Figure 2
Thermal ellipsoid (30%) drawing of Ar′CrCrAr′ (Ar′ means C6H3-2,6(C6H3-2,6-Pri2)2, where Pri means isopropyl). Hydrogen atoms are not shown. Selected bond distances and angles (deg) are as follows: Cr(1)–Cr(1A), 1.8351(4) Å; Cr(1)–C(1), 2.131(1) Å; Cr(1)–C(7A), 2.2943(9) Å; Cr(1)–C(8A), 2.479(1) Å; Cr(1)–Cr(12A), 2.414(1) Å; C(1)–C(2), 1.421(1) Å; C(1)–C(6), 1.423(2) Å; C(7)–C(8), 1.421(1) Å; C(7)–C(12), 1.424(1) Å; Cr(1A)–Cr(1)–C(1), 108.78(3)°; Cr(1A)–Cr(1)–C(7A), 94.13(3)°; C(1)–Cr(1)–C(7A), 163.00(4)°; Cr(1)–C(1)–C(2), 114.34(7)°; Cr(1)–C(1)–C(6), 131.74(7)°; and C(2)–C(1)–C(6), 113.91(9)°. Reprinted with permission from ref (40). Copyright 2005 The American Association for the Advancement of Science.
Figure 3
Figure 3
Structures of Schrock and Clark’s “yl-ene-yne” complex of refs (41, 42), Wilkinson’s nitride-imido complex (half of the dimer is shown and the chloride bridges to one lithium of a chemically equivalent manganese center) of ref (46), and the NCr(NPh) (NPri2)2 anion of ref (43). Reprinted with permission from ref (43). Copyright 2016 PCCP Owner Societies.
Figure 4
Figure 4
MADs of the ccCA-TM composite method with respect to experimental heats of formation. Unit is kcal/mol along the Y axis. The value in brackets represents the number of data points in the subset. Reprinted with permission from ref (47). Copyright 2011 American Chemical Society.
Figure 5
Figure 5
Complex [Cu(AcHG1G2G3NH2)(Py)(W)] used as a precursor of all of the five- and four-coordinated forms considered in the study of Bruschi et al. (A) Schematic representation of the complex; (B) molecular geometry of the complex alone; and (C) the complex inserted into a sphere of 84 water molecules. Reprinted with permission from ref (290). Copyright 2012 American Chemical Society.
Figure 6
Figure 6
Accuracy of bond distances (upper panel) and bond angles (lower panel) in complexes of calcium with various organic functional groups reproduced by the DFTB3 parametrization from Kubillus et al. The deviations observed with PBE/def2-SVP calculations are shown for comparison. The reference data were obtained with DFT optimizations at the level B3LYP/def2-TZVPP using Turbomole 6.4. Errors expressed as MADs in pm. The test systems contained in each category mentioned here are detailed further in the Supporting Information of ref (293). Reprinted with permission from ref (293). Copyright 2015 American Chemical Society.
Figure 7
Figure 7
MADs of the bond distances of Mg2+ and Zn2+ test sets in ref (294) using the PM6, DFTB3/MIO, DFTB3/3OB, MP2/aug-cc-pVTZ, and B3LYP/6-31+G(d,p) methods where B3LYP/aug-cc-pVTZ is used as reference. Reprinted with permission from ref (294). Copyright 2014 American Chemical Society.
Figure 8
Figure 8
Simulated XRD patterns of the Cu-BTC MOF with the experimental unit cell parameters and optimized at the DFTB and DFT levels of theory. Reprinted with permission from ref (297). Copyright 2011 John Wiley and Sons.
Figure 9
Figure 9
Jacob’s ladder of density functional approximations. Any resemblance to the Tower of Babel is purely coincidental. Also shown are angels in the spherical approximation, ascending and descending. Users are free to choose the rungs appropriate to their accuracy requirements and computational resources. However, at present (ca. 2001), their safety can be guaranteed only on the two lowest rungs. Reprinted with permission from ref (302). Copyright 2001 American Institute of Physics Publishing LLC.
Figure 10
Figure 10
Potential energy curve for Cr2 computed with the mPW exchange functional and the KCIS correlation function using the following percentages of HF exchange: X = 0 (×), 1 (+), 2 (◇), 3 (□), 4 (△), 5 (○), 6 (*), 7 (−), and 8 (●), and experiment (solid line). Reprinted with permission from ref (311). Copyright 2005 American Chemical Society.
Figure 11
Figure 11
Optimized bond lengths for the LSDA (△), GGA (×), hybrid GGA (□), meta GGA (◇), and hybrid meta GGA (○) with TZQ basis level and the experimental bond length (line) for Cr2. Reprinted with permission from ref (311). Copyright 2005 American Chemical Society.
Figure 12
Figure 12
Multiple electronic configurations that originate from the electronic interaction of two VII ions. Reprinted with permission from ref (310). Copyright 2012 American Chemical Society.
Figure 13
Figure 13
MAPEs for the interatomic distances, δ; cohesive energies, Ecoh; and bulk moduli, B0, of 27 transition metals with respect to experimental values extrapolated to 0 K and adjusted to remove zero-point vibrational contributions. MAPEs of δ has been multiplied by a factor of 10 for a better presentation. Data for LDA xc functionals, PBE, PW91, PBEsol, and RPBE are adapted from Janthon et al. Reprinted with permission from ref (325). Copyright 2012 American Chemical Society.
Figure 14
Figure 14
RMSDs of PBE versus percentage of HF exchange for 3d species. Reprinted with permission from ref (317). Copyright 2014 American Chemical Society.
Figure 15
Figure 15
RMSDs of PBE versus percentage of HF exchange for 4d species. Reprinted with permission from ref (317). Copyright 2014 American Chemical Society.
Figure 16
Figure 16
Average unsigned heat of formation errors (kcal/mol) for the complete set of complexes, highlighting the performance of different M0X suite of functionals with LANL2DZ basis set, containing transition metals taken in the study of Shil et al. Reprinted with permission from ref (318). Copyright 2013 American Chemical Society.
Figure 17
Figure 17
Average unsigned heat of formation errors (kcal/mol) for different transition metal bonding partners treated in the study of Shil et al. Reprinted with permission from ref (318). Copyright 2013 American Chemical Society.
Figure 18
Figure 18
MADs of 22 DFT functionals and the rp-ccCA method for the HOFs of 30 molecules containing 4d TMs. Reprinted with permission from ref (314). Copyright 2013 American Chemical Society.
Figure 19
Figure 19
Representation of Rmin,ij, σij, and εij in a 12-6 LJ equation. Here, we illustrate the LJ potential between two particles, which have the same Rmin,i (1.6 Å) and εi (1.1 kcal/mol) values.
Figure 20
Figure 20
Illustration of the difference between SLEF and the conventional 1/r Coulomb function in describing charge interactions between Zn2+ and the oxygen of TIP3P water: (A) energy and (B) force. The parameters in the SLEF1 force field were employed. Reprinted with permission from ref (479). Copyright 2011 American Chemical Society.
Figure 21
Figure 21
Model system [Zn(Et)2(Im)2]0 representing the ZnCys2His2 binding structure used in fitting the CT and binding energy as a function of the distance between zinc and the ligated atom. Reprinted with permission from ref (481). Copyright 2013 American Chemical Society.
Figure 22
Figure 22
QM calculated CT (Δqct, NPA charge) at the MP2/6-311++g(2df,p) level as a function of the coordinate bond length for ethylthiolate in ZnCys2His2, ZnCys4, and ZnCys3His1 systems. Solid lines denote analytical fitted curves of QM results. Reprinted with permission from ref (481). Copyright 2013 American Chemical Society.
Figure 23
Figure 23
Interaction energies of ethylthiolate (Et) and [Zn(Et)(Im)2]+ as a function of coordinate bond length. Eqm is the binding energy of Et and the [Zn(Et)(Im)2]+ calculated by Gaussian 09 at the MP2/6-311++g(2df,p) level. Eele is the electrostatic interaction between Et and the [Zn(Et)(Im)2]+. Evdw is the van der Waals interaction energy between Et and the [Zn(Et)(Im)2]+, and Ect is “charge-transfer” energy between Et and the [Zn(Et)(Im)2]+ (Ect = EqmEeleEvdw). Reprinted with permission from ref (481). Copyright 2013 American Chemical Society.
Figure 24
Figure 24
Atomic ions that were parametrized for the 12-6 model by Li, Merz, and co-workers.,, These ions are shown with a blue background. The monovalent anions, monovalent cations, divalent cations, trivalent cations, and tetravalent cations are shown in light green, yellow, orange, red, and dark red, respectively. The ions that have multiple oxidation states are shown in white, which are Cr2+/Cr3+, Fe2+/Fe3+, Cu+/Cu2+, Ag+/Ag2+, Ce3+/Ce4+, Sm2+/Sm3+, Eu2+/Eu3+, and Tl+/Tl3+. Besides these atomic ions, the H3O+ and NH4+ ions were also parametrized in their work. Reprinted with permission from the Ph.D. dissertation of Li. Copyright 2016 Pengfei Li.
Figure 25
Figure 25
Determination of the three parameter sets for the Zn2+ ion in TIP3P. Reprinted with permission from ref (390). Copyright 2013 American Chemical Society.
Figure 26
Figure 26
Osmotic pressure as a function of the NaCl (a) or KCl (b) salt concentration. The red line is the experimental values; green line is obtained from MD simulations with the LB combination rules; blue line and magenta line are from simulations with adjusted Rmin LJ parameters using NBFIX. In panel a, the blue line is the original Na+ (1) from ref (540), and the magenta line is the reparametrized Na+ (2) from ref (541). The error bars are calculated from 10 separate trajectories of 1 ns. Reprinted with permission from ref (539). Copyright 2010 American Chemical Society.
Figure 27
Figure 27
HFE and IOD fitting curves of six representative divalent metal ions using TIP3P water model. Reprinted with permission from ref (390). Copyright 2013 American Chemical Society.
Figure 28
Figure 28
Fitting curves between the HFE and IOD values for the positive (left) and negative (right) monovalent ions with three water models together with the targeted values of the ions investigated in the work of Li et al. Reprinted with permission from ref (510). Copyright 2015 American Chemical Society.
Figure 29
Figure 29
HFE absolute errors (left) and percentage errors (right) for the IOD parameter sets for the mono-, di-, tri-, and tetravalent cations in the TIP3P water model. The ratio is roughly 1.0:3.0:4.8:14.2 for the errors in the HFE values for the mono-, di-, tri-, and tetravalent cations. This agreed well with the ion-induced dipole interaction (eq 59) that predicts that strength of the ion-induced dipole is approximately proportional to the square of the ion’s charge (ratio of 12:22:32:42). Reprinted with permission from ref (510). Copyright 2015 American Chemical Society.
Figure 30
Figure 30
Illustration of the reparametrization scheme used in the work of Yoo and Aksimentiev. Oxygen is colored in red, magnesium in pink, carbon in cyan, phosphorus in tan, hydrogen in white, and salt ions in yellow. Reprinted with permission from ref (544). Copyright 2012 American Chemical Society.
Figure 31
Figure 31
Atomic ions that were parametrized for the 12-6-4 model by Li, Merz, and co-workers.,, These ions are shown with a blue background. The monovalent anions, monovalent cations, divalent cations, trivalent cations, and tetravalent cations are shown in light green, yellow, orange, red, and dark red, respectively. The ions that have multiple oxidation states are shown in white, which are Cr2+/Cr3+, Fe2+/Fe3+, Cu+/Cu2+, Ce3+/Ce4+, and Tl+/Tl3+. Besides these atomic ions, the H3O+ and NH4+ ions were also parametrized in their work. Reprinted with permission from the Ph.D. dissertation of Li. Copyright 2016 Pengfei Li.
Figure 32
Figure 32
Scheme representing intermolecular interactions: the green double-headed arrow and red double-headed arrow represent the interactions, which are included and not included in the 12-6 nonbonded model, respectively. Reprinted with permission from ref (549). Copyright 2014 American Chemical Society.
Figure 33
Figure 33
Available force fields and challenges in transferability. (a) Parameters for different energy expressions can hardly processed into a uniform force field for both organic and inorganic components. (b) Harmonic force fields like AMBER, CHARMM, OPLS-AA, and PCFF are similar and contain quality parameters for organic compounds. Bonded terms and atomic charges for newly added compounds are derived in similar manners. Moderate differences in the type of LJ potential, combining rules, and the scaling of nonbonding interactions between 1,4-bonded atoms (1,4 nonbonded scaling) can be overcome by adjustments in σ0,ii and ε0,ii to reproduce the same bulk and interfacial properties. Reprinted with permission from ref (598). Copyright 2013 American Chemical Society.
Figure 34
Figure 34
Cu+ binding site from the HAH1 crystal structure, including Cys side chains, and the three QM models used to represent two-, three-, and four-coordinate Cu+ environments. Reprinted with permission from ref (654). Copyright 2007 American Chemical Society.
Figure 35
Figure 35
Proposed mechanism for transfer of Cu+ between HAH1 and MNK4. A possible four-coordinate intermediate that would reside between the two three-coordinate states in the middle has been left out. In the HAH1 homodimer used as a mimic of the HAH1-MNK4 heterodimer in MD simulations, C14 and C17 of the target domain are replaced by Cys 12 B and Cys 15 B, respectively. Reprinted with permission from ref (654). Copyright 2007 American Chemical Society.
Figure 36
Figure 36
Initial structure of K. aerogenes urease in the open state, as used in the simulations of ref (659). The trimeric subunits are shown in yellow, pink, and cyan. The flaps are shown as α-helices (blue) and loops (red). Nickel ions are shown as green spheres. Reprinted with permission from ref (659). Copyright 2012 American Chemical Society.
Figure 37
Figure 37
Heavy atoms of the metal site residues in native KA urease. The two nickel ions in the metal center are six-coordinated (left one) and five-coordinated (right one), respectively. The structure is from PDB entry 1FWJ, and this figure was made using VMD.
Figure 38
Figure 38
Flap, active site, and ancillary binding pocket of K. aerogenes urease. The flap is shown in yellow, the active site in red, and the ancillary binding pocket in blue. The nickel ions are shown as green spheres, and the exposed nickel surfaces in (D)–(I) are also shown in green. The three rows from top to bottom show the closed flap (A–C), the open flap (D–F), and the wide-open flap (G–I). In each row, three views are shown: from the top, looking down into the binding pockets (A, D, G); from the side (B, E, H); and from the side in a cutaway view (C, F, I). To show the scale, a urea molecule (4 Å across) is depicted at the bottom of the figure. Reprinted with permission from ref (659). Copyright 2012 American Chemical Society.
Figure 39
Figure 39
Relative free energy map for opening of the urease flap. The points labeled as “C”, “O”, and “W”, respectively, represent the closed, open, and wide-open structures shown in Figure 38. Points C and O represent the initial XRD structures of the closed- and open-flap models (PDB entries 1FWJ and 1EJX, respectively); for the latter structure, the flap was added by homology modeling. Point W is a representative wide-open conformation selected from the simulation trajectory of 1FWJ. Reprinted with permission from ref (659). Copyright 2012 American Chemical Society.
Figure 40
Figure 40
Four allosteric forms of CzrA examined in the study of Chakravorty et al., and the associated thermodynamic parameters of the CzrA mechanism. The protein, CzrA, is shown in red, Zn2+ ions are in green, and the DNA molecules are blue. These cartoon depictions are snapshots from their MD simulations of these allosteric forms. A negative cooperative allosteric effect of ∼6 kcal/mol is observed on Zn2+ binding to CzrA. Reprinted with permission from ref (656). Copyright 2012 American Chemical Society.
Figure 41
Figure 41
Snapshot of the CzrA metal binding site from the initial structure. The zinc ion is represented as a silver sphere. The metal binding residues include Asp84, His86, His97′, and His100′. These residues also represent the QM region of the QM/MM MD calculations in the study of Chakravorty et al. Reprinted with permission from ref (656). Copyright 2012 American Chemical Society.
Figure 42
Figure 42
Plot of RMSF values for the lowest mode of the protein from simulation of Apo·CzrA starting from the apo-crystal structure, and simulation of allosteric switching for Zn·CzrA starting from the 2KJB NMR structure of apo-CzrA in a closed conformation. Reprinted with permission from ref (656). Copyright 2012 American Chemical Society.
Figure 43
Figure 43
Metal ions currently supported by the MCPB.py program. The metals with a blue background use the VDW parameters from Li et al.,,, whereas the metals with a green background use the VDW parameters adapted from UFF. Reprinted with permission from ref (671). Copyright 2016 American Chemical Society.
Figure 44
Figure 44
(a) Bond length versus bond force constant; blue ◆ are bonds with hydrogen atoms. (b) Angle versus angular force constant; the red ■ represents the angle between carbon atoms in octanitrocubane. Adapted with permission from ref (663). Copyright 2011 Springer.
Figure 45
Figure 45
Linear regression for the main ligands in Mn-coordination spheres: aspartate, glutamate, histidine, and water, shown in red, orange, blue, and green, respectively. Reprinted with permission from ref (641). Copyright 2013 American Chemical Society.
Figure 46
Figure 46
Second-order polynomial regression for the main donor atoms in Mn-coordination spheres (oxygen and nitrogen are shown in red and blue colors, respectively). Reprinted with permission from ref (641). Copyright 2013 American Chemical Society.
Figure 47
Figure 47
Average equilibrium angles and force constants for main donor atoms concerning the equilibrium angles for the main geometries in manganese. Reprinted with permission from ref (641). Copyright 2013 American Chemical Society.
Figure 48
Figure 48
Newman projections depicting viable Zn(CH3)2 conformations. Reprinted with permission from ref (205). Copyright 2009 American Chemical Society.
Figure 49
Figure 49
Relative conformational energy versus H–C–C–H twist for 1Zn(CH3)2 (CCSD/6-31G**); unrelaxed scan in 5° increments. Minima at −120°, 0°, and 120° correspond to the pseudoeclipsed conformation; maxima at −60°, 60°, and 180° correspond to pseudogauche. Reprinted with permission from ref (205). Copyright 2009 American Chemical Society.
Figure 50
Figure 50
Deformation map in peptide bond plane. (A) Static map obtained from database parameters. (B) Experimental static map obtained using atomic charges and multipole parameters refined against the crambin diffraction data. (C) Theoretical static map computed for the pseudomonopeptide (Z)-N-acetyl-α,β-dehydrophenylalanine methylamide. (D) Experimental dynamic map for the peptide Ala-9-Arg-10 of crambin. The contour level is 0.05 e3. Positive, red lines; negative, blue lines. Reprinted with permission from ref (759). Copyright 2000 National Academy of Sciences, U.S.A.
Figure 51
Figure 51
Extended Born model for the example of silicon dioxide (SiO2). Single-headed arrows describe reversibly taken steps, as indicated in the text of ref (752). Partial ionization energies and electron affinities are involved in step 3. The LE is divided into an electrostatic/repulsive component in step 4 plus a covalent component in step 5. The respective stoichiometric coefficients must be remembered. The shaded, double-headed arrows indicate the complementary electrostatic (top) and covalent (bottom) contributions to the cycle. Reprinted with permission from ref (752). Copyright 2004 American Chemical Society.
Figure 52
Figure 52
Most important criteria for obtaining thermodynamically consistent force field parameters. (a) Atomic charges and (b) computed interfacial properties in the models should quantitatively agree with atomic-scale and macroscale experimental data. The color scheme and single-headed arrows emphasize the feed of information from experiment (blue) into models and theory (green). Reprinted with permission from ref (598). Copyright 2013 American Chemical Society.
Figure 53
Figure 53
(a) Ball-and-stick representation of the atoms of the unit cell of DMOF-1 (Zn, O, N, C, and H atoms are represented as light blue, red, dark blue, gray, and white atoms, respectively). (b) Cluster created by cutting directly a piece of framework. This cluster cannot be used to model the environment of the BDC (1,4-benzenedicarboxylate) ligand and calculate its charges, because there are cleaved bonds that will have very different electronic structures than in the bulk structure. (c) Same cluster shown in (b), although the cleaved bonds have been saturated with H atoms to achieve electronic structures in the terminal N and C atoms that are similar to those in the crystal structure. Reprinted with permission from ref (768). Copyright 2015 Elsevier.
Figure 54
Figure 54
Definition of CBAC atom types and their charges. Reprinted with permission from ref (773). Copyright 2010 American Chemical Society.
Figure 55
Figure 55
(a) Cluster used for calculating the atomic partial charges for atoms V, O9, and O10. (b) Comparison of simulated and experimental CO2 excess adsorption isotherm in MIL-47(V) (V, green; O, red; C, gray, H, white; Li, purple). Reprinted with permission from ref (773). Copyright 2010 American Chemical Society.
Figure 56
Figure 56
Surface of the least-squares fits (∑Δ2) of atomic LJ parameters ε and r0 (which is σ in eq 33) for Pt(II) complexes interacting with the testing molecules. Reprinted with permission from ref (775). Copyright 2016 American Chemical Society.
Figure 57
Figure 57
Geometry of the cationic dummy atom model with octehdral shape and a charge of n+, in which the central metal ion with a charge of n – 6δ bonds covalently to six dummy sites each with a charge of +δ. Reprinted with permission from ref (786). Copyright 2016 American Chemical Society.
Figure 58
Figure 58
(a) Diagram showing the effects of metal ion and general-base catalysts on the reaction of eq 105. The abscissa denotes increasing general-base strength, represented by a water molecule, a carboxylate ion, and an imidazole ring. The ordinate represents increasing metal ion electrophilicity, where a water molecule denotes the case where no metal is present. The energy values, ΔGij, correspond to the proton-transfer reaction (M2+i)H2O + Bj ⇌ (M2+i)OH + BH+j, where each entry is obtained from ΔGij = −2.3RT(pKj – pKi). For example, in the case of a metalloenzyme using Ca2+ and glutamate, respectively, as M and B, i denotes Ca2+(H2O) and j denotes (Glu)-COOH (the pKa’s for metal bound water are taken from ref (789)). (b) The figure shows a number of different enzymes that catalyze reactions involving a proton-transfer step like eq 105, plotted according to their use of metal ion and general-base catalysis (SNase, DNase I, and RNase denote staphylococcal nuclease, deoxyribonuclease I, and ribonucleases [e.g., A and T1], respectively). Reprinted with permission from ref (785). Copyright 1990 American Chemical Society.
Figure 59
Figure 59
Illustration of the multisite ion model for calcium in the study of Saxena and Sept. Instead of a simple sphere, the charge centers are now distributed to multiple sites depending on the coordination geometry of the ion. Reprinted with permission from ref (792). Copyright 2013 American Chemical Society.
Figure 60
Figure 60
Structure of the minimum corresponding to 12 water molecules interacting with the Cr3+ hydrate using the HIWP and the MCHO potentials in the study of Martínez et al. Drawn bonds between the cation and oxygens of the first-shell water molecules represent the feature of the HIWP to consider the hydrate as a single unit. Reprinted with permission from ref (803). Copyright 2000 American Institute of Physics Publishing LLC.
Figure 61
Figure 61
Correlation diagram between the absolute electronegativity parameters and the ABEEM parameters χ* for some metal ions. Reprinted with permission from ref (848). Copyright 2007 American Chemical Society.
Figure 62
Figure 62
Correlation diagram between the absolute hardness parameters and the ABEEM parameters 2η* for some metal ions. Reprinted with permission from ref (848). Copyright 2007 American Chemical Society.
Figure 63
Figure 63
Schemes of the DO and DR models for Au particles. Reprinted with permission from ref (884). Copyright 2008 John Wiley and Sons.
Figure 64
Figure 64
Using the real gold atoms as LJ interaction site drives adsorption in the hollow position (left panel), while the use of VS (in blue) drives adsorption on-top (right panel). Reprinted with permission from ref (513). Copyright 2008 John Wiley and Sons.
Figure 65
Figure 65
Lateral view of the adsorption geometry obtained by DFT calculations for all 14 molecules in Table 1 of ref (513). Only the uppermost gold layer is shown in the figure. Reprinted with permission from ref (513). Copyright 2008 John Wiley and Sons.
Figure 66
Figure 66
Pyrimidine dicarboxamide inhibitors binding with MMP-13. Reprinted with permission from ref (936). Copyright 2012 American Chemical Society.
Figure 67
Figure 67
Comparison of calculated and experimental relative binding free energies (kcal/mol). Ligands 4, 3, 2, and 1 can be identified from the x axis in order from left to right, respectively, with the corresponding experimental binding free energies −11.05, −9.75, −8.73, and −7.07 kcal/mol. Reprinted with permission from ref (936). Copyright 2012 American Chemical Society.
Figure 68
Figure 68
Crystal structure (light color) of the Zif268 Cys2His2 Zn-finger domain superimposed upon the average MD structure (dark color) derived from simulations using (a) the CTPOL force field, and (b) and (c) two 12-6 LJ nonbonded models. The metal-binding site is in black (MD) or gray (X-ray), while the regular secondary structures are in dark blue (MD) and light blue (X-ray), whereas the loops are in red (MD) and pink (X-ray). Reprinted with permission from ref (924). Copyright 2005 American Chemical Society.
Figure 69
Figure 69
Crystal structure of the Zn-Cys4 adenylate kinase lid domain superimposed upon the average MD structure derived from simulations. Panel a is for the CTPOL force field, while panels b and c are for two 12-6 LJ nonbonded models. The metal-binding site is in black (MD) or gray (X-ray), while the regular secondary structures are in dark blue (MD) and light blue (X-ray), whereas the loops are in red (MD) and pink (X-ray). Reprinted with permission from ref (924). Copyright 2005 American Chemical Society.
Figure 70
Figure 70
X-ray structure (panel a), average structure of MD simulation (panel b), superimposition of metal site of the former two structures (panel c), and RMSD values of the Cα atoms in protein backbone, Cd-Cys4 complex, and average Zn-Cys4 complex (solid, gray, and dotted curves in panel d, respectively). Reprinted with permission from ref (955). Copyright 2008 John Wiley and Sons.
Figure 71
Figure 71
Nonpolar (NP), polarization (Pol.), and polarization plus charge-transfer (Pol+CT) energies in the energy decomposition of ion–water dimer system using the FQ-DCT model. Na+–water, Cl–water, Mg2+–water, and Zn2+–water correspond to panels a, b, c, and d, respectively. Reprinted with permission from ref (555). Copyright 2015 American Chemical Society.
Figure 72
Figure 72
CSOV (blue) versus SIBFA (red) electrostatic (top left), repulsion (top right), polarization (bottom left), and charge-transfer (bottom right) energies as a function of the Th–O distance in Th4+–H2O. Reprinted with permission from ref (935). Copyright 2012 Springer.
Figure 73
Figure 73
Interatomic distance dependency of the carbon–carbon bond order in ReaxFF. Reprinted with permission from ref (987). Copyright 2001 American Chemical Society.
Figure 74
Figure 74
(a) Uncorrected bond orders for carbon and hydrogen within an ethane molecule. Carbon is overcoordinated due to second nearest neighbor interactions with hydrogen atoms. (b) Corrected bond orders. Weak interactions are negated due to carbon having its full bond order of 4. Reprinted with permission from ref (168). Copyright 2011 Elsevier.
Figure 75
Figure 75
Plot of the Coulombic and VDW energies versus distance. A shielding term is applied to prevent these energies from becoming too large. Reprinted with permission from ref (168). Copyright 2011 Elsevier.
Figure 76
Figure 76
Comparison of the ReaxFF calculated charges versus QM values for the same molecule. The ReaxFF closely reproduces the Mullikan charges on each atom and models the polarizability of the molecule. Reprinted with permission from ref (168). Copyright 2011 Elsevier.
Figure 77
Figure 77
Product distribution observed during the NVT-MD simulation of CH3OH exposed to V2O5 (001) using ReaxFF. Only major products are shown explicitly; minor intermediates are included as “others”, and carbon-containing intermediates bound to the V2O5 surface are included as “surface carbon species”. Reprinted with permission from ref (1002). Copyright 2008 American Chemical Society.
Figure 78
Figure 78
Configurations obtained after 90 ps of NVT-MD simulation using ReaxFF without metal (a) and with 15 Co (b), Ni (c), and Cu (d). Reprinted with permission from ref (994). Copyright 2005 American Chemical Society.
Figure 79
Figure 79
Configuration obtained from the NVT-MD simulation using ReaxFF with 15 Ni atoms after 750 ps. Reprinted with permission from ref (994). Copyright 2005 American Chemical Society.
Figure 80
Figure 80
Cu–O RDF and integral as obtained from a ReaxFF MD simulation on a [Cu(H2O)216]2+ system at T = 300 K. Reprinted with permission from ref (1008). Copyright 2010 American Chemical Society.
Figure 81
Figure 81
Overlay of LFMM (yellow) and DFT (blue) structures: top left, [Cu(SMe)4]2–, top right, [Cu(SMe2)4]2+, and bottom, the Type I active site model [Cu(im)2(SMe)(SMe2)]+. Reprinted with permission from ref (1023). Copyright 2006 Royal Society of Chemistry.
Figure 82
Figure 82
Overlay of XRD (yellow) and LFMM (blue) structures. Figures on the left show the superposition of the protein backbones, while those on the right show the optimal overlay of the trigonal [CuNNS] set. The proteins are given in the same order as Table 1 of ref (1023). Reprinted with permission from ref (1023). Copyright 2006 Royal Society of Chemistry.
Figure 83
Figure 83
Schematic diagrams of Fe2+ am(m)ine complexes. In the solid state, 1 and 2 are high-spin while 3 and 4 are low-spin. Reprinted with permission from ref (1024). Copyright 2010 American Chemical Society.
Figure 84
Figure 84
Comparison of theoretical spin-state energy differences (kJ/mol) for complexes 1–4. Reprinted with permission from ref (1024). Copyright 2010 American Chemical Society.
Figure 85
Figure 85
Comparison of PESs for angular distortion of water calculated by (●) the RHF method with a 6-31G* basis set, (○) a harmonic force field (k = 0.191 hartree/rad2, θeq = 105.5°), and (■) a Fourier term (kF = 0.0652 hartree, n = 2.42, φ = 74.50°). Reprinted with permission from ref (197). Copyright 1991 American Chemical Society.
Figure 86
Figure 86
Representation of the skeletal modes of a linear metallocene. The Cp rings are represented by horizontal lines and the metal by a dark circle. The 1,5-dihedral angle, α, the M–D stretch, a, the D–M–D, β, and the C–D–M bend, γ, are shown. Reprinted with permission from ref (1020). Copyright 1992 American Chemical Society.
Figure 87
Figure 87
Strength functions for sp3, sp2, sp1, and pure p orbitals. Reprinted with permission from ref (1012). Copyright 1993 American Chemical Society.
Figure 88
Figure 88
Scheme of π-bonding p–d atomic orbitals in the VALBOND model. Reprinted with permission from ref (1015). Copyright 2001 American Chemical Society.
Figure 89
Figure 89
Overlap representation between a dπ orbital (left) and a σ bond with sdn hybridization (right). Reprinted with permission from ref (1015). Copyright 2001 American Chemical Society.
Figure 90
Figure 90
Representation of a cloverleaf shaped orbital by two dz2 shaped orbitals. Reprinted with permission from ref (1015). Copyright 2001 American Chemical Society.
Figure 91
Figure 91
Minimized structure of Cu2+(H2O)4(H2O)2 based on the SIBFA (panel A) and SIBFA-LF (panel B) models. Reprinted with permission from ref (1018). Copyright 2003 John Wiley and Sons.
Figure 92
Figure 92
Minimized structure of Cu2+(ImH)4 based on the SIBFA (panel A) and SIBFA-LF (panel B) models. Reprinted with permission from ref (1018). Copyright 2003 John Wiley and Sons.
Figure 93
Figure 93
Schematic plot of VB angular potential for each 3c4e bond based on 10% s and 90% d hybridization. Reprinted with permission from ref (1017). Copyright 2012 John Wiley and Sons.
Figure 94
Figure 94
Energy difference between square-planar (sq) and tetrahedral (te) tetra-aqua TM complexes; energy calculated by (UsqUsq/empty) – (UteUte/empty); data points from AMOEBA and AMOEBA-VB methods for [Zn(H2O)4]2+ overlap each other, as the differences in results are very small; see Tables 3 and 4 in the Supporting Information of ref (1017) for numerical values. Usq/empty means system energy without considering the central ion. Reprinted with permission from ref (1017). Copyright 2012 John Wiley and Sons.
Figure 95
Figure 95
Potential energy difference between square-planar and tetrahedral tetra-aqua Cu2+ complexes with the water–water interaction removed. Negative values indicate that the square-planar structure is lower in potential energy than the tetrahedral geometry. Reprinted with permission from ref (1016). Copyright 2014 American Chemical Society.

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