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Theoretical Approaches to Chemical Reactivities • I Potential Energy Surfaces (PES) • II Use of PES in the Study of Chemical Phenomena • III High Accuracy (Theoretical) Thermochemistry -Master in Physical and Theoretical Chemistry Quy Nhon University, 25 – 31 March 2013 by Minh Tho NGUYEN, K.U.Leuven, Belgium Born-Oppenheimer Approximation • HΨ = EΨ (1) • H(r,R) = TN(R) + Te(r) + VNe(r,R) + Vee(r) + VNN (R) (2) WIth fixed nuclei, the electronic motion can be described by: • [Te(r) + Vee(r) + VNe(r,R) + VNN(R)] φ(r,R) = εn(R) φn(r,R) • Total molecular WF: ) Ψ (r , R= ∑χ n (3) ( R )Φ n ( r , R ) n  Adiabatic Representation is based on the fact that a nuclear mass is >1800 times as large as the electron mass; nuclei move much slower than electrons • One term WF: Ψ (r,R) = χn(R) φn(r,R) • Define an Adiabatic Halmiltonian: Hn(adia) = TN(R) + εn(R) + Λnn(R) • The nuclear wavefunction: Hn (adia) χn (R) = E χn (R) - The validity of the adiabatic approximation resides in the assumption that the nuclear kinetic energy is substantially smaller than the energy gap between adiabatic electronic states The non-adiabatic coupling arising from the nuclear kinetic energy could be considered as a perturbation of the total electronic energy and as a consequence, the nuclei should contain a small amount of kinetic energy for a slow motion - Electronic eigenfunction is real, the coupling terms are small and can be removed  Born-Oppenheimer Approximation: [TN(R) + Vn(R) ] χn(R) = En χn(R) • The electronic energy Vn(R) plays the role of potential for the nuclear motions • This motion occurs on a potential energy surface: a molecular system undergoes a chemical reaction when its nuclei move smoothly on a PES which is nothing else than its electronic energies • The nuclear motion from one to another region induces a reorganisation of its electronic density to adapt to the novel nuclear configuration, but not the transition to other electronic states • although both terms are often used to describe the separation of electronic motion from that of the nuclear motion, there is a difference between the adiabatic approximation and BO approximation • while the BO PES is independent of the nuclear masses, the diagonal coupling terms in the Adiabatic approximation depend on the latter • To achieve high accuracy, the coupling term must be retained In view of the practical difficulties in evaluating the nonadiabatic coupling terms, non-Born-Oppenheimer calculations are presently achievable for atoms and small diatomic molecules • In a perturbation treatment of the BOA, the first-order correction to the BO electronic energy due to the nuclear motion is the Diagonal BO Corrections (DBOC) • in the DBOC, only the diagonal terms are computed from EDBOC = which thus depends on the nuclear mass  Small at the chemical energy scale but very relevant for a spectroscopic energy scale ( ~100 cm-1) In some cases, DBOC needs to be added to obtain accurate heats of formation Features of Potential Energy Surfaces  The total energy E of a molecular system is calculated as a function of internal coordinates qk: E = E(qk) k from to 3N-6  Search for stationary points where energy gradients (first derivatives of energy) vanish gk = δE / δqk k = … 3N-6  With energy Hessians (second derivatives), local minima (equilibrium structures) and first-order saddle points (transition structures) can be located and characterized Hkl = δ2E / δqk δql k, l = …… 3N-6  All Eigenvalues positive: λk >  minimum One negative eigenvalue: λk <  first-order saddle-point • Statement 1:  An equilibrium structure corresponds to an energy minimum • Statement 2: A transition structure for a chemical process for a reaction must be a first-order saddle point; a saddle-point that is a maximum in only one direction and a minimum in all other perpendicular directions Reaction mechanism via TS structures Potential Energy (Eelec+ Vnn) Transition Structure (TS) Reactants Products Reaction Coordinate • Statement 3: A Reaction Coordinate should be considered as a combination of internal coordinates  In practice, only portions of a potential energy surface can be mapped out, and relevant energy profiles established  A minimum energy reaction pathway is determined by the intrinsic reaction coordinate (IRC) The path of steepest descent from a saddle-point is not unique, because the shape of a PES depends on the particular choice of coordinates that describes the geometry More than one combination of bond lengths, bond angles and dihedral angles can be employed to represent the same structure Expl: - NH3 inversion - a 1,2-H-shift - cycloaddition  Vibrations Energy Surfaces • Harmonic oscillator approximation of motions on a PES: E(q) = E0 + ½ Σij Fij (qi – qi0) (qj – qj 0) Fij = δ2E / δqi δqj = force constants The harmonic vibrational frequencies for a molecule can be computed from the force constant matrix, the geometry and the atomic masses by solution of the Wilson-GF method: det (GF – εI) = where G-1 is the kinetic energy matrix (geometry + masses) and eigenvalues ε are related to the vibrational frequencies: ω = 1/2π √ ε The normal coordinates of vibration are those linear combinations of the coordinates for which the GF matrix equation can be solved In massweighted coordinates (G being constant times the unit matrix), the normal coordinates are just the eigenvectors of F “Guessing” a TS Geometry • Base the guess on a previously calculated, related system, or ‘chemical intuition’ and a preconceived notion of the mechanism • Use an ‘average’ of the reactant and product geometries: Linear Synchronous Transit method) • a Quadratic Synchronous Transit method, in which minima perpendicular to the LST are connected • Don’t be disappointed if optimizations fail: several attempts are needed! Locating TS • Perform a low level calculation (HF, B3LYP/3-21G or /631G*) • Because TS’s involve partial bonding, lower levels of theory are not always useful • If found, use that result as starting point for higher level calculation, with analytical frequencies at the first point • Verify with a frequency calculation at the same high level of theory and basis set • When failed: change the geometry, basis set, method … • In shallow surface, or coupled coordinates, analytical Hessians at every point may be needed to have the right curvature Confirming a Possible TS • Must be a first order saddle point on PES smoothly connecting reactant to product – Verify that the Hessian yields one and only one negative (imaginary) frequency – Animate the normal coordinate corresponding to the imaginary frequency; it should connect reactants and products (have vibrations consistent with expected bond breaking and bond forming) – It is imperative to run the IRC pathway when not sure about the identity of the minima • Confirm the TS by different levels of theory (HF, MP2, DFT basis sets ) Locating a TS: • Surface Fitting: the simplest way is to fit an analytical expression to computed energies Problems: 1.Acceptable functional form must be foundfor multidimension surface A large number a energy should be computed The fitted surface may not be sufficiently accurate in the region of the barrier to provide an acceptable estimate of the TS geometry Analytical form is needed in chemical dynamics and trajectory calculations, in conformation analysis, or the crossing of ground and excited states • Linear Synchronous Transit (LST): Assume that the reaction path is a straight line connecting reactants and products  the LST TS is always higher than the true TS From this, one can minimize the energy w.r.t all coordinates perpendicular to the linear path The resulting point is lower in energy than the true TS The reaction path can now be described by a parabola or quadratic curve that connect the minima quadratic synchronous transit (QST) The new maximum is a better estimate , and can be repeated until the TS is found LST and QST Approaches • Advantage: only energy, no gradient, and less costly than a surface fitting • Linear interpolation of distance matrices  gives a reasonable pathway • Problems when the path is strongly curve, and the reactants and products are very different • In any case, when using a low-level method, this give a quick scan of the surface to get indication on the shape • The estimate TS can certainly be valubale for optimization using other methods • Locating a TS: Coordinate Driving: In many reaction, the change of one coordinate dominates the region of the surface,the TS can be located by following this coordinate Problem encountered when the path should be curved so that a second coordinate begins to dominate (e.g 1,2-H-shift): the method fails giving a discontinuity on the surface, or to lead to a dead-end valley  at best, a supperposition of many single coordinate curves could help identify the saddle region • Walking up Valleys: A simple approach to improve the coordinate driving method is to take a step of a specific length and then determine the direction that corresponds to the shallowest path up the valley Eigenvector Following Method (EF): In following the appropriate vector of the Hessian at each step Hij needs to be recalculated frequently  high cost If a guess starts with a negative eigenvalue of the Hessian and the transition vector is corresponding, and it remains negative, even the magnitude and the direction could be changed, the EF method usually leads to a proper TS after a limited number of steps If the Hessians are available (from calculations at a lower level), and a guess with a clear direction, the EF method represents a reliable method for finding TSs In constrat to the minima, there is no guaranty that the process converges to a TS Symmetry and Stationary Points • In favorable circumstances, symmetry can be used to turn a TS optimization into a minimum location • Constraint could lead to faster convergence avoiding the involvement of lower frequency modes • If a higher symmetry structure is suspected, it is often more efficient to optimize the higher symmetry structure directly and test it w.r.t distortion to lower symmetry (HP=NH) Choice of coordinates • Faster convergence if the coordinate system is chosen with some care Strong coupling between coordinates invariably causes difficulties for optimizations (stiff = stretch + bend; loose = internal rotation + inversion; cycles, linear = inherent strong coupling and also redundant coordinates) • Coupling of the transition vector with the low frequency mode leads to problem for TSs • Loosely bound clusters: flat surafce with large geometrical changes may occur with small energy changes •  Use of internal coordinates with dummy atoms (5rings, cycloadditions) Summary for TS searching • Coordinate Driving, or potential energy surface scan (1D) or small grid (2D)  locate the transition region • LST – QST • Once a reasonable guess is obtained, and the gradient and the hessians can be coumputed / estimated, the reaction coordinate may be frozen in a minimization Then the eigenvector following method with g and H represents the best strategy • The problem is to update the Hessians In some difficult cases, recalculation of the full H at each iteration may be necessary even it is the most expensive one What to when optimization fails Forces too large: error in the input: poor geometry, or bad coordinate system  reconstruct the Z-matrix too avoid strong coupling Negative eigenvalue during an optimization: structure is not a minimum or numerical problem with Hessian updates  follow the negative eigenvector; or improve the Hij If this persists, just impose the symmetry, or freeze the relevant coordinate When the structure is optimized, release the frozen coordinate and reoptimize Too many negative eigenvalues:  reoptimize following the lower frequency vector, or update for better Hessians No negative eigenvalue during a TS optimization: bad geometry guess or bad Hessian Eigenvalues too large  error in input, or bad Hessian Eigenvalues too small  shallow minimum, or redundancy (more than 3N-6 coordinates), or Hessian are not good updated  tighten the convergence criteria (RMS gradient) Number of steps exceeded: difficult case, reoptimize rather than increase the number Don’t go beyond 50 iterations, it won’t converge Change in point group detected during an optimization: z-matrix does not reflect the full symmetry, or optimization inadvertently distorts the molecule  use the Cartesian coordinates Danger of Computational Science Garbage.in Black Box Garbage.out Users are fully and solely responsible for the accuracy of the data and their interpretations