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Lecture6

The MIT 10.637 Lecture 6 covers transition state searches and potential energy surface (PES) sampling methods, focusing on finding saddle points, which are more challenging than minima. Various techniques such as quadratic synchronous transit (QST), nudged elastic band (NEB), and string methods are discussed for optimizing transition states and minimizing energy paths, emphasizing the importance of coordinate selection. Additionally, dynamic methods like umbrella sampling and steered molecular dynamics are introduced to effectively sample high-energy regions and enhance probability distributions during simulations.

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MIT 10.637 Lecture 6 Transition state searches and PES sampling hjkulik@mit.edu September 23, 2014
MIT 10.637 Lecture 6 N > 3, General PES features from H. B. Schlegel, Wayne State U. We focus on a reaction coordinate defined by critical DOF, not full 3N-6!
MIT 10.637 Lecture 6 Energy minimizations Ñf = ¶f ¶x1 , , ¶f ¶xn æ è ç ö ø ÷ Ñf = 0 Force = -Ñ(Energy)Minimize: Gradient on our PES in terms of all coordinates (internal, cartesian): For : Any stationary point (a) local minimum, (b) global minimum, (c) saddle point. What we usually want! What we want today!
MIT 10.637 Lecture 6 Finding saddle points • Saddle points are more difficult to find than minima. • Interpolation methods: assume reactant and product minima are known and that a TS is located “between” these two end-points. May only find a geometry very close to the TS, rather than the true TS. • Local methods: propagate geometry using information about the function, first and second derivatives – do not need to know reactant/product but usually need to have a good estimate of the TS. Once TS is found, can get reaction path by following steepest descent to R and P from TS.
MIT 10.637 Lecture 6 Saddle points in small systems For extremely small systems: PES is 3N-6 variables, so it may be possible to map out the complete PES in limited cases. For simple small/intermediate systems: “coordinate driving” or constrained optimizations. For cases where there are a few internal degrees of freedom that describe the difference between reactant and product: e.g. torsion angle, bond distances for breaking/forming. Choose fixed values for selected coordinates and remaining variables are optimized, adiabatically mapping energy as f(reaction variables). TS is geometry where residual gradients for fixed variables are “sufficiently small”.
MIT 10.637 Lecture 6 Saddle points in small systems Coordinate driving: Reaction variables are good if they have large coefficients in the Hessian eigenvector with negative eigenvalue at the TS. BUT we only know the Hessian of the TS after it has been found, making the reaction coordinate selection very use-rbiased. If only one or two variables change significantly between Reactants and products, this method works well, e.g. Rotation of a methyl group – torsion angle HNC to HCN – HCN angle SN2 reactions X +CH3Y  XCH3 + Y – XC and CY distances (pictured) More than two reaction coordinates becomes difficult and rarely is successful.
MIT 10.637 Lecture 6 Synchronous transit Linear synchronous transit: can consider this a coordinate driving method. All degrees of freedom – Cartesian or internal – are varied linearly between reactant and product. No optimization is performed. Assumptions: 1) all variables change at the same rate along the reaction path. 2) transition state is the highest energy structure along the interpolation line. However, assuming all variables changing the same amount all at the same time is a bad approximation. This method rarely leads to a good estimate of the TS.
MIT 10.637 Lecture 6 Synchronous transit R TS P Quadratic synchronous transit (QST): Approximate reaction path by a parabola instead of a straight line. Find the maximum on the LST, generate QST by minimizing energy perpendicular to the LST path. Search for maximum on QST path.
MIT 10.637 Lecture 6 The Hessian/Force matrix (3N-6)x(3N-6) matrix with elements: Hij ( f ) = ¶2 f ¶xi¶xj H( f ) = ¶2 f ¶x1 2 ¶2 f ¶x1¶x2 ¶2 f ¶x1¶xn ¶2 f ¶x2¶x1 ¶2 f ¶x2 2 ¶2 f ¶x2¶xn ¶2 f ¶xn¶x1 ¶2 f ¶xn¶x2 ¶2 f ¶xn 2 é ë ê ê ê ê ê ê ê ê ê ê ù û ú ú ú ú ú ú ú ú ú ú (3N-6) (3N-6) Approximate PES around stationary point by harmonic potentials.
MIT 10.637 Lecture 6 The Hessian/Force matrix Diagonalize Hessian (eigenvalue problem) From E. Eliav, Tel Aviv U minimum maximum saddle point eklj (k) = Hijli (k) i=1 n å ek = mwk 2 Eigenvalues: Eigenvectors: normal coordinates Harmonic frequencies εk>0 εk<0 εk>0, except one εj<0 ωk all real ωk all imaginary one imaginary ωj on RC
MIT 10.637 Lecture 6 Eigenvector following • In the vicinity of transition state, can use information from second derivative (Hessian) matrix. • Can be expensive to compute all second derivatives for large systems (3N-6x3N-6) so some method to compute only elements needed or to approximate can speed up approach. • Need good initial guess for the transition state. • Works best if Hessian already has only one negative eigenvalue – (e.g. augmented-Hessian NR) and need actual Hessian and not just initial approximation! Expensive!
MIT 10.637 Lecture 6 Synchronous transit Improving on LST and QST: Another class of methods starts with QST solution and then follows eigenvector to the saddle point – “guided QST” Introduced by Schlegel and coworkers, Synchronous Transit-guided Quasi- Newton (STQN) methods: -Can use parabola or circle arc for interpolation -Then use tangent to guide search towards the TS. -Once TS region is located, optimization is switched to quasi-Newton-Raphson -Can require either just reactants and products or reactants, transition, state and products. Note: these methods generally all work for simple systems and coordinates: the interpolation may be nonsensical in Cartesians but meaningful in internals, etc.
MIT 10.637 Lecture 6 Rational function optimization Rational function optimization: Recall from geometry optimizations lecture – RFO expands the function in terms of a rational approximation and then have to solve for a shift parameter that makes eigenvalues positive. The quadratic approximation – requires step length to be equal to the trust radius also. Partial rational function optimization (P-RFO) for TS searches: Now use two shift parameters l and lTS: Choose lTS such that search will maximize the energy in this direction, but step can be outside trust radius. QA or “TRIM” uses l=-lTS :
MIT 10.637 Lecture 6 Problems with eigenvector following • Need one eigenvector weakly coupled to others to follow (i.e. small 3rd order derivatives). • NR may fail to converge if the eigenvectors are too coupled and there’s not just one eigenvalue to follow. • Need a good geometry to start from. • Hessian is expensive. NR TS search requires explicit Hessian from the start.
MIT 10.637 Lecture 6 The dimer method Eliminates the need to calculate the Hessian: Dimer is oriented towards lowest curvature mode by rotation and translation. Curvature is estimated by finite difference from only the forces on the two images, avoiding the need for a Hessian calculation. Once the dimer is aligned on the lowest curvature mode, force parallel to the dimer is inverted from the total force, making the dimer climb up to the saddle point:
MIT 10.637 Lecture 6 Nudged elastic band Chain-of-states method: string of interpolated images/geometries is propagated. This is a modification of the elastic band (Elber & Karplus) method. A chain gang initial state final state guesses (reactants) (products) Springs keep interpolated images separated: Our chain of states are propagated on the potential energy surface until we find a minimum energy path. Typically 4-20 images
MIT 10.637 Lecture 6 Nudged elastic band Nudged elastic band is a method that propagates a series of interpolated images towards the minimum energy path. Initial interpolations of the images are usually linear – and if done in Cartesian coordinates (common), one has to be careful as there can be clashing/overlap of atoms. Spring forces keep the images spaced equally. Otherwise they’d fall downhill to the reactant and product states. We apply the spring force along the tangent (i.e. along the path): i-1 i i+1 Spring constant Distance i+1 to i Distance i to i-1 Tangent unit vector
MIT 10.637 Lecture 6 Nudged elastic band i-1 i i+1 The energy is minimized perpendicular to the path – i.e. forces along the path are projected out and replaced with the spring force. The perpendicular force is the true force minus the component that is parallel to the path: The total force in NEB is the sum of the spring force along the tangent and the true force perpendicular to the tangent:
MIT 10.637 Lecture 6 Nudged elastic band Mueller potential
MIT 10.637 Lecture 6 Nudged elastic band Mueller potential Reactants, intermediates, products Saddle point
MIT 10.637 Lecture 6 Nudged elastic band Mueller potential Reactants, intermediates, products Saddle point minimum energy path ÑE Ri( )^ = 0
MIT 10.637 Lecture 6 Nudged elastic band Mueller potential Reactants, intermediates, products Saddle point minimum energy path NEB initial guess from interpolation a la LST
MIT 10.637 Lecture 6 Nudged elastic band Mueller potential Reactants, intermediates, products Saddle point minimum energy path NEB initial guess from interpolation NEB image i Fi NEB Fi S Fi ^ Fi ˆti
MIT 10.637 Lecture 6 Nudged elastic band i Fi NEB Fi S Fi ^ Fi ˆti Fi NEB = Fi ^ +Fi S Fi ^ = - ÑE Ri( )-ÑE Ri( )× ˆti( ) Fi S = k Ri+1 -Ri - Ri -Ri-1( ) ˆti Spring forces: only want component of this force that keeps images separated (along path). True forces: ignore component that minimizes energy parallel to path, only minimize perpendicularly. NEB image force:
MIT 10.637 Lecture 6 Nudged elastic band Mueller potential Reactants, intermediates, products Converged NEB path NEB image
MIT 10.637 Lecture 6 Nudged elastic band The climbing image extension to nudged elastic band ensures that an image converges to the saddle point. While NEB guarantees images to be equally spaced, no guarantee that there’s an image at the top of the barrier – in fact, it’s very unlikely. Select the highest energy image and invert the force: This maximum energy image is not affected by the spring forces. Qualitatively: this image moves up the PES along the band and down in energy perpendicular to the band. Need to have enough images close to the CI to get a good estimate of the path. Total force Force parallel to the tangent
MIT 10.637 Lecture 6 Nudged elastic band Variable spring constants have been introduced to ensure convergence of the most important points along the minimum energy path – around the saddle point. This is extra important when climbing image is used because you need to have a good estimate of the tangent to the path near the saddle point. Choose strong forces near the saddle point and weak forces further away. The force constants are linearly scaled based on the energy of the image: Higher energy endpoint
MIT 10.637 Lecture 6 Nudged elastic band Example of climbing image results vs. standard NEB results Example of variable springs results vs. standard NEB results Reaction coordinate RelativeEnergy Variable springs Fixed springs Reaction coordinate RelativeEnergy Climbing image Standard
MIT 10.637 Lecture 6 Nudged elastic band NEB paths can be difficult to converge: when the force parallel to the MEP is large compared to the force perpendicular, when there are inflection points. Kinks can form on the band and oscillate. Improved tangent method revises the definition of the tangent from: A slight improvement ensures equispaced images even in areas of large curvature: LEPS potential + harmonic oscillator with NEB path in dashed line showing kinks
MIT 10.637 Lecture 6 Nudged elastic band Adding in perpendicular spring force when the angle between the vectors between neighboring images (Ri-Ri-1 vs Ri+1-Ri) deviated from zero helps. However, this leads to corner cutting on the MEP in curved regions. If the saddle point is in a curved region, then the saddle point energy will be overestimated. Improved tangent method: Use only the tangent to the image that’s higher in energy: Special case where image I is at maximum or minimum:
MIT 10.637 Lecture 6 Nudged elastic band Modified tangent NEB converges on the LEPS+H.O. potential Modified tangent smooths out convergence. Can also show that there’s a stability criterion for kinking in the path. The following must be satisfied: F < 2CR Where F is the perpendicular force to the path and C is the curvature around the MEP. Path will always become unstable for enough images because R goes down for large # images.
MIT 10.637 Lecture 6 String method The string method finds a minimum energy path (f*) connecting points A+B. The MEP satisfies the constraint: The force perpendicular to the path is zero. The string method is similar to the NEB method – it propagates a path of images, e.g. where a is the normalized arc length between A and B. There’s a constraint that the parameterization is preserved when the string deforms or the local arc length is constant along the string: or This means that the elastic energy in the string is distributed evenly.
MIT 10.637 Lecture 6 String method Using the constraint developed for the string along with the definition of the MEP, we can rewrite the expression for the MEP: Where t is the tangent vector and l is a Langrange multiplier that imposes the constraint. In order to propagate a string towards the MEP is to carry out steepest descent dynamics in the string space: The expression for NEB written in the same terminology is:
MIT 10.637 Lecture 6 String method Can propagate the string with damped dynamics approach instead to ensure rapid convergence: where gs is a friction coefficient. Can propagate these with Verlet algorithm: Where we have a modified friction factor and force: The values of the Lagrange multipliers can be solved iteratively, based on the fact that the distance between images is constant and the expression for propagation.
MIT 10.637 Lecture 6 String method • Main difference between NEB and string is that the images get repositioned at every iteration in the string method. • Extensions to the string method include: – Growing string method: start two strings – one from reactants and one from products. Grow them until they join and then propagate towards MEP. – Frozen string method: Same as growing string but once images are added, they are frozen. More efficient than growing string. – Finite temperature string: incorporates temperature effects.
MIT 10.637 Lecture 6 Further considerations • Good coordinate choices are very important for transition state optimization • Good coordinate choice can enlarge the convergence region, reducing requirement of a good starting geometry. Poor coordinates, decrease convergence region. • Eigenvector following methods work for stiff systems – small eigenvalues in Hessians will be a problem. • Small eigenvalues/soft modes in the Hessian are better suited to path-based methods. • Transition state may be a minimum for a certain symmetry (vs. the symmetry of the reactants and products).
MIT 10.637 Lecture 6 Dynamic free energy methods • For large systems, it may be difficult to optimize transition states via the methods we’ve discussed so far. • Instead, may want to carry out simulation dynamically. • There’s also the consideration of free energy (G), which has been excluded so far – only have looked at enthalpy (H).
MIT 10.637 Lecture 6 Umbrella sampling Direct dynamics will not sample high-energy regions near the TS if there’s an activation energy. Need to bias the sampling, use a penalty to the potential to force sampling of high energy regions: The biasing potential can be made to be sufficiently steep, so the energy far from r0 is very high in energy. Then will only sample near r0. This method is known as umbrella sampling. The ensemble is not Boltzmann but it can be deconvoluted.
MIT 10.637 Lecture 6 Umbrella sampling Reaction coordinate Probability Umbrella sampling locally enhances probability in low- probability regions by altering the potential. Perform many biased simulations at different positions along reaction path, calculate free energy – called Potential of Mean Force: this is done by unweighting and stitching together the underlying free energy function. Weighted histogram-analysis method (WHAM) is a common approach to stitching together the free energy: deconvolute contribution of force constant from curvature of underlying free energy surface.
MIT 10.637 Lecture 6 Steered molecular dynamics Steered molecular dynamics is a technique where constant force or velocity is applied to select atoms to move them from one configuration to another. A set of collective variables is chosen to define progress along a reaction coordinate, e.g. proton transfer in malonaldehyde: Collective variable: dist(1)-dist(2). Once reaction coordinate is propagated, can get PMF: Guiding potential: Work: PMF: (1) (2) May need to run several (hundreds?) of SMD runs to get accurate PMF.
MIT 10.637 Lecture 6 Metadynamics Introduced in 2002 by Laio and Parrinello, metadynamics is a way to speed up dynamics and sample free energy landscapes of complex systems. Collective variables (up to 3, typically 2) are selected to define the free energy landscape, e.g. coordination number of relevant species participating in a reaction, bond distances or angles, etc. The most challenging aspect of metadynamics is choosing the right collective variables. A biasing Gaussian potential is added to the real potential: New Gaussian is added at every tG in space of collective variables, S are the coordinates, s is the collective variable, w is the height of the Gaussian, ds is the width of the Gaussian.
MIT 10.637 Lecture 6 Metadynamics The potential wells are gradually filled up minima until the full landscape is traversed. The free energy surface is obtained by taking the negative image of all of the Gaussians. If Gaussians are too large or added too frequently, free energy surface will be inaccurate, Gaussians too small or infrequently added – will take too long for simulation to finish.
MIT 10.637 Lecture 6 Summary • For small systems with simple reaction coordinates: coordinate driving, linear synchronous transit, quadratic synchronous transit, or synchronous transit- guided-Quasi-Newton can be good approaches. • For intermediate systems: string method or nudged elastic band • For large systems with many DOF or to compute free energies: steered MD, metadynamics, umbrella sampling • Coordinate selection and initial guess for a reaction pathway are almost always challenging.
MIT 10.637 Lecture 6 Survey! bit.ly/lec-6
MIT 10.637 Lecture 6 Transition state theory Reaction coordinate Energy E a kBT Reactants Products Transition State Activation energy is one piece of the puzzle – full exploration of TST allows us to calculate absolute rates, through prefactor calculations. A potential energy surface connects the reactants and products: this is a QM concept! The activated complex or transition state is at the maximum potential energy. Once configuration is achieved, fast decay to products. Concentration of the transition state is dependent upon equilibrium with reactants, vibration along the path. 45
MIT 10.637 Lecture 6 The Eyring equation Entropy of activation term Enthalpy of activation term Macroscopic Eyring equation: Microscopic Eyring equation: These are the molecular partition functions of A and B and the activated complex, AB. The Eyring equation is a rigorous method of obtaining kinetics from free energy of activation and may be derived from statistical mechanics (not here):
MIT 10.637 Lecture 6 The Eyring equation and TST In the derivation of the Eyring equation (not shown), one vibrational frequency is stripped out of the molecular partition function for the activated complex. This is assumed to be the mode that carries the complex over the barrier to the products. Also: note that TST assumes once the activated complex is formed, it readily proceeds to products and does not cross back, which can be inaccurate.
MIT 10.637 Lecture 6 Interpretation of 48 The potential energy surface is represented by changes in the internal coordinates of the molecule(s). These correspond to vibrational modes that alter internal coordinates or translation of the reactants with respect to each other. At the TS, we have one “molecule” and motion is characterized by the vibrational modes. One imaginary vibrational frequency characterizes the transition state and correlates to the attempt frequency in our derivation of the Eyring equation. Imaginary frequency of the transition state for HCNHNC isomerization
MIT 10.637 Lecture 6 Partition function review # Maxwell-Boltzmann E Fraction molecules in state i. b= 1/kBT energy of the state Boltzmann distribution Molecular partition function sum over j states: sum over i levels: 1 2 3 4 5 6 1 2 3 4 5 6g=5 g=1
MIT 10.637 Lecture 6 Molecular partition function contributions are derived from solving eigenvalue problems in quantum mechanics.
MIT 10.637 Lecture 6 Molecular degrees of freedom Linear Nonlinear Translational 3 3 Rotational 2 3 Vibrational 3N-5 3N-6 For zero field, potential energy derived only from vibrational coordinates. e.g. H2O: 3 normal modes symmetric stretch asymmetric stretchbend
MIT 10.637 Lecture 6 Vibrational zero-point energy n=0 n=1 n=2 1 2 ~! 3 2 ~! 5 2 ~! From Q.M., molecules have zero point energy: ZPE Reaction coordinate + B P AB‡ A E0 Ea Energy Enthalpy of activation:
MIT 10.637 Lecture 6 Contributions to prefactors DOF form f(TX) if if Order ½ ½ 0 1 1 10 108 cm-1 53
MIT 10.637 Lecture 6 Prefactors from quantum chemistry • Must have structures that are at minima or saddle points – vibrationally characterize them to get zero point energy and vibrational partition function. • ZPE corrects the enthalpy of activation. • Rotational partition function may also be calculated and its contribution is added to the partition functions. • Electronic terms are neglected. • Translational terms only depend on temperature.
MIT 10.637 Lecture 6 Summary • It is possible to use a saddle point structure and minimum energy structures to obtain absolute rates of reaction by calculating prefactors from statistical mechanics. • However, relative rates, where the prefactors are assumed to be the same for different reactions are typically preferred unless the method is extremely accurate.

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Lecture6

  • 1.
    MIT 10.637 Lecture 6 Transition statesearches and PES sampling hjkulik@mit.edu September 23, 2014
  • 2.
    MIT 10.637 Lecture 6 N >3, General PES features from H. B. Schlegel, Wayne State U. We focus on a reaction coordinate defined by critical DOF, not full 3N-6!
  • 3.
    MIT 10.637 Lecture 6 Energy minimizations Ñf= ¶f ¶x1 , , ¶f ¶xn æ è ç ö ø ÷ Ñf = 0 Force = -Ñ(Energy)Minimize: Gradient on our PES in terms of all coordinates (internal, cartesian): For : Any stationary point (a) local minimum, (b) global minimum, (c) saddle point. What we usually want! What we want today!
  • 4.
    MIT 10.637 Lecture 6 Finding saddlepoints • Saddle points are more difficult to find than minima. • Interpolation methods: assume reactant and product minima are known and that a TS is located “between” these two end-points. May only find a geometry very close to the TS, rather than the true TS. • Local methods: propagate geometry using information about the function, first and second derivatives – do not need to know reactant/product but usually need to have a good estimate of the TS. Once TS is found, can get reaction path by following steepest descent to R and P from TS.
  • 5.
    MIT 10.637 Lecture 6 Saddle pointsin small systems For extremely small systems: PES is 3N-6 variables, so it may be possible to map out the complete PES in limited cases. For simple small/intermediate systems: “coordinate driving” or constrained optimizations. For cases where there are a few internal degrees of freedom that describe the difference between reactant and product: e.g. torsion angle, bond distances for breaking/forming. Choose fixed values for selected coordinates and remaining variables are optimized, adiabatically mapping energy as f(reaction variables). TS is geometry where residual gradients for fixed variables are “sufficiently small”.
  • 6.
    MIT 10.637 Lecture 6 Saddle pointsin small systems Coordinate driving: Reaction variables are good if they have large coefficients in the Hessian eigenvector with negative eigenvalue at the TS. BUT we only know the Hessian of the TS after it has been found, making the reaction coordinate selection very use-rbiased. If only one or two variables change significantly between Reactants and products, this method works well, e.g. Rotation of a methyl group – torsion angle HNC to HCN – HCN angle SN2 reactions X +CH3Y  XCH3 + Y – XC and CY distances (pictured) More than two reaction coordinates becomes difficult and rarely is successful.
  • 7.
    MIT 10.637 Lecture 6 Synchronous transit Linearsynchronous transit: can consider this a coordinate driving method. All degrees of freedom – Cartesian or internal – are varied linearly between reactant and product. No optimization is performed. Assumptions: 1) all variables change at the same rate along the reaction path. 2) transition state is the highest energy structure along the interpolation line. However, assuming all variables changing the same amount all at the same time is a bad approximation. This method rarely leads to a good estimate of the TS.
  • 8.
    MIT 10.637 Lecture 6 Synchronous transit R TS P Quadraticsynchronous transit (QST): Approximate reaction path by a parabola instead of a straight line. Find the maximum on the LST, generate QST by minimizing energy perpendicular to the LST path. Search for maximum on QST path.
  • 9.
    MIT 10.637 Lecture 6 The Hessian/Forcematrix (3N-6)x(3N-6) matrix with elements: Hij ( f ) = ¶2 f ¶xi¶xj H( f ) = ¶2 f ¶x1 2 ¶2 f ¶x1¶x2 ¶2 f ¶x1¶xn ¶2 f ¶x2¶x1 ¶2 f ¶x2 2 ¶2 f ¶x2¶xn ¶2 f ¶xn¶x1 ¶2 f ¶xn¶x2 ¶2 f ¶xn 2 é ë ê ê ê ê ê ê ê ê ê ê ù û ú ú ú ú ú ú ú ú ú ú (3N-6) (3N-6) Approximate PES around stationary point by harmonic potentials.
  • 10.
    MIT 10.637 Lecture 6 The Hessian/Forcematrix Diagonalize Hessian (eigenvalue problem) From E. Eliav, Tel Aviv U minimum maximum saddle point eklj (k) = Hijli (k) i=1 n å ek = mwk 2 Eigenvalues: Eigenvectors: normal coordinates Harmonic frequencies εk>0 εk<0 εk>0, except one εj<0 ωk all real ωk all imaginary one imaginary ωj on RC
  • 11.
    MIT 10.637 Lecture 6 Eigenvector following •In the vicinity of transition state, can use information from second derivative (Hessian) matrix. • Can be expensive to compute all second derivatives for large systems (3N-6x3N-6) so some method to compute only elements needed or to approximate can speed up approach. • Need good initial guess for the transition state. • Works best if Hessian already has only one negative eigenvalue – (e.g. augmented-Hessian NR) and need actual Hessian and not just initial approximation! Expensive!
  • 12.
    MIT 10.637 Lecture 6 Synchronous transit Improvingon LST and QST: Another class of methods starts with QST solution and then follows eigenvector to the saddle point – “guided QST” Introduced by Schlegel and coworkers, Synchronous Transit-guided Quasi- Newton (STQN) methods: -Can use parabola or circle arc for interpolation -Then use tangent to guide search towards the TS. -Once TS region is located, optimization is switched to quasi-Newton-Raphson -Can require either just reactants and products or reactants, transition, state and products. Note: these methods generally all work for simple systems and coordinates: the interpolation may be nonsensical in Cartesians but meaningful in internals, etc.
  • 13.
    MIT 10.637 Lecture 6 Rational functionoptimization Rational function optimization: Recall from geometry optimizations lecture – RFO expands the function in terms of a rational approximation and then have to solve for a shift parameter that makes eigenvalues positive. The quadratic approximation – requires step length to be equal to the trust radius also. Partial rational function optimization (P-RFO) for TS searches: Now use two shift parameters l and lTS: Choose lTS such that search will maximize the energy in this direction, but step can be outside trust radius. QA or “TRIM” uses l=-lTS :
  • 14.
    MIT 10.637 Lecture 6 Problems witheigenvector following • Need one eigenvector weakly coupled to others to follow (i.e. small 3rd order derivatives). • NR may fail to converge if the eigenvectors are too coupled and there’s not just one eigenvalue to follow. • Need a good geometry to start from. • Hessian is expensive. NR TS search requires explicit Hessian from the start.
  • 15.
    MIT 10.637 Lecture 6 The dimermethod Eliminates the need to calculate the Hessian: Dimer is oriented towards lowest curvature mode by rotation and translation. Curvature is estimated by finite difference from only the forces on the two images, avoiding the need for a Hessian calculation. Once the dimer is aligned on the lowest curvature mode, force parallel to the dimer is inverted from the total force, making the dimer climb up to the saddle point:
  • 16.
    MIT 10.637 Lecture 6 Nudged elasticband Chain-of-states method: string of interpolated images/geometries is propagated. This is a modification of the elastic band (Elber & Karplus) method. A chain gang initial state final state guesses (reactants) (products) Springs keep interpolated images separated: Our chain of states are propagated on the potential energy surface until we find a minimum energy path. Typically 4-20 images
  • 17.
    MIT 10.637 Lecture 6 Nudged elasticband Nudged elastic band is a method that propagates a series of interpolated images towards the minimum energy path. Initial interpolations of the images are usually linear – and if done in Cartesian coordinates (common), one has to be careful as there can be clashing/overlap of atoms. Spring forces keep the images spaced equally. Otherwise they’d fall downhill to the reactant and product states. We apply the spring force along the tangent (i.e. along the path): i-1 i i+1 Spring constant Distance i+1 to i Distance i to i-1 Tangent unit vector
  • 18.
    MIT 10.637 Lecture 6 Nudged elasticband i-1 i i+1 The energy is minimized perpendicular to the path – i.e. forces along the path are projected out and replaced with the spring force. The perpendicular force is the true force minus the component that is parallel to the path: The total force in NEB is the sum of the spring force along the tangent and the true force perpendicular to the tangent:
  • 19.
  • 20.
    MIT 10.637 Lecture 6 Nudged elasticband Mueller potential Reactants, intermediates, products Saddle point
  • 21.
    MIT 10.637 Lecture 6 Nudged elasticband Mueller potential Reactants, intermediates, products Saddle point minimum energy path ÑE Ri( )^ = 0
  • 22.
    MIT 10.637 Lecture 6 Nudged elasticband Mueller potential Reactants, intermediates, products Saddle point minimum energy path NEB initial guess from interpolation a la LST
  • 23.
    MIT 10.637 Lecture 6 Nudged elasticband Mueller potential Reactants, intermediates, products Saddle point minimum energy path NEB initial guess from interpolation NEB image i Fi NEB Fi S Fi ^ Fi ˆti
  • 24.
    MIT 10.637 Lecture 6 Nudged elasticband i Fi NEB Fi S Fi ^ Fi ˆti Fi NEB = Fi ^ +Fi S Fi ^ = - ÑE Ri( )-ÑE Ri( )× ˆti( ) Fi S = k Ri+1 -Ri - Ri -Ri-1( ) ˆti Spring forces: only want component of this force that keeps images separated (along path). True forces: ignore component that minimizes energy parallel to path, only minimize perpendicularly. NEB image force:
  • 25.
    MIT 10.637 Lecture 6 Nudged elasticband Mueller potential Reactants, intermediates, products Converged NEB path NEB image
  • 26.
    MIT 10.637 Lecture 6 Nudged elasticband The climbing image extension to nudged elastic band ensures that an image converges to the saddle point. While NEB guarantees images to be equally spaced, no guarantee that there’s an image at the top of the barrier – in fact, it’s very unlikely. Select the highest energy image and invert the force: This maximum energy image is not affected by the spring forces. Qualitatively: this image moves up the PES along the band and down in energy perpendicular to the band. Need to have enough images close to the CI to get a good estimate of the path. Total force Force parallel to the tangent
  • 27.
    MIT 10.637 Lecture 6 Nudged elasticband Variable spring constants have been introduced to ensure convergence of the most important points along the minimum energy path – around the saddle point. This is extra important when climbing image is used because you need to have a good estimate of the tangent to the path near the saddle point. Choose strong forces near the saddle point and weak forces further away. The force constants are linearly scaled based on the energy of the image: Higher energy endpoint
  • 28.
    MIT 10.637 Lecture 6 Nudged elasticband Example of climbing image results vs. standard NEB results Example of variable springs results vs. standard NEB results Reaction coordinate RelativeEnergy Variable springs Fixed springs Reaction coordinate RelativeEnergy Climbing image Standard
  • 29.
    MIT 10.637 Lecture 6 Nudged elasticband NEB paths can be difficult to converge: when the force parallel to the MEP is large compared to the force perpendicular, when there are inflection points. Kinks can form on the band and oscillate. Improved tangent method revises the definition of the tangent from: A slight improvement ensures equispaced images even in areas of large curvature: LEPS potential + harmonic oscillator with NEB path in dashed line showing kinks
  • 30.
    MIT 10.637 Lecture 6 Nudged elasticband Adding in perpendicular spring force when the angle between the vectors between neighboring images (Ri-Ri-1 vs Ri+1-Ri) deviated from zero helps. However, this leads to corner cutting on the MEP in curved regions. If the saddle point is in a curved region, then the saddle point energy will be overestimated. Improved tangent method: Use only the tangent to the image that’s higher in energy: Special case where image I is at maximum or minimum:
  • 31.
    MIT 10.637 Lecture 6 Nudged elasticband Modified tangent NEB converges on the LEPS+H.O. potential Modified tangent smooths out convergence. Can also show that there’s a stability criterion for kinking in the path. The following must be satisfied: F < 2CR Where F is the perpendicular force to the path and C is the curvature around the MEP. Path will always become unstable for enough images because R goes down for large # images.
  • 32.
    MIT 10.637 Lecture 6 String method Thestring method finds a minimum energy path (f*) connecting points A+B. The MEP satisfies the constraint: The force perpendicular to the path is zero. The string method is similar to the NEB method – it propagates a path of images, e.g. where a is the normalized arc length between A and B. There’s a constraint that the parameterization is preserved when the string deforms or the local arc length is constant along the string: or This means that the elastic energy in the string is distributed evenly.
  • 33.
    MIT 10.637 Lecture 6 String method Usingthe constraint developed for the string along with the definition of the MEP, we can rewrite the expression for the MEP: Where t is the tangent vector and l is a Langrange multiplier that imposes the constraint. In order to propagate a string towards the MEP is to carry out steepest descent dynamics in the string space: The expression for NEB written in the same terminology is:
  • 34.
    MIT 10.637 Lecture 6 String method Canpropagate the string with damped dynamics approach instead to ensure rapid convergence: where gs is a friction coefficient. Can propagate these with Verlet algorithm: Where we have a modified friction factor and force: The values of the Lagrange multipliers can be solved iteratively, based on the fact that the distance between images is constant and the expression for propagation.
  • 35.
    MIT 10.637 Lecture 6 String method •Main difference between NEB and string is that the images get repositioned at every iteration in the string method. • Extensions to the string method include: – Growing string method: start two strings – one from reactants and one from products. Grow them until they join and then propagate towards MEP. – Frozen string method: Same as growing string but once images are added, they are frozen. More efficient than growing string. – Finite temperature string: incorporates temperature effects.
  • 36.
    MIT 10.637 Lecture 6 Further considerations •Good coordinate choices are very important for transition state optimization • Good coordinate choice can enlarge the convergence region, reducing requirement of a good starting geometry. Poor coordinates, decrease convergence region. • Eigenvector following methods work for stiff systems – small eigenvalues in Hessians will be a problem. • Small eigenvalues/soft modes in the Hessian are better suited to path-based methods. • Transition state may be a minimum for a certain symmetry (vs. the symmetry of the reactants and products).
  • 37.
    MIT 10.637 Lecture 6 Dynamic freeenergy methods • For large systems, it may be difficult to optimize transition states via the methods we’ve discussed so far. • Instead, may want to carry out simulation dynamically. • There’s also the consideration of free energy (G), which has been excluded so far – only have looked at enthalpy (H).
  • 38.
    MIT 10.637 Lecture 6 Umbrella sampling Directdynamics will not sample high-energy regions near the TS if there’s an activation energy. Need to bias the sampling, use a penalty to the potential to force sampling of high energy regions: The biasing potential can be made to be sufficiently steep, so the energy far from r0 is very high in energy. Then will only sample near r0. This method is known as umbrella sampling. The ensemble is not Boltzmann but it can be deconvoluted.
  • 39.
    MIT 10.637 Lecture 6 Umbrella sampling Reactioncoordinate Probability Umbrella sampling locally enhances probability in low- probability regions by altering the potential. Perform many biased simulations at different positions along reaction path, calculate free energy – called Potential of Mean Force: this is done by unweighting and stitching together the underlying free energy function. Weighted histogram-analysis method (WHAM) is a common approach to stitching together the free energy: deconvolute contribution of force constant from curvature of underlying free energy surface.
  • 40.
    MIT 10.637 Lecture 6 Steered moleculardynamics Steered molecular dynamics is a technique where constant force or velocity is applied to select atoms to move them from one configuration to another. A set of collective variables is chosen to define progress along a reaction coordinate, e.g. proton transfer in malonaldehyde: Collective variable: dist(1)-dist(2). Once reaction coordinate is propagated, can get PMF: Guiding potential: Work: PMF: (1) (2) May need to run several (hundreds?) of SMD runs to get accurate PMF.
  • 41.
    MIT 10.637 Lecture 6 Metadynamics Introduced in2002 by Laio and Parrinello, metadynamics is a way to speed up dynamics and sample free energy landscapes of complex systems. Collective variables (up to 3, typically 2) are selected to define the free energy landscape, e.g. coordination number of relevant species participating in a reaction, bond distances or angles, etc. The most challenging aspect of metadynamics is choosing the right collective variables. A biasing Gaussian potential is added to the real potential: New Gaussian is added at every tG in space of collective variables, S are the coordinates, s is the collective variable, w is the height of the Gaussian, ds is the width of the Gaussian.
  • 42.
    MIT 10.637 Lecture 6 Metadynamics The potentialwells are gradually filled up minima until the full landscape is traversed. The free energy surface is obtained by taking the negative image of all of the Gaussians. If Gaussians are too large or added too frequently, free energy surface will be inaccurate, Gaussians too small or infrequently added – will take too long for simulation to finish.
  • 43.
    MIT 10.637 Lecture 6 Summary • Forsmall systems with simple reaction coordinates: coordinate driving, linear synchronous transit, quadratic synchronous transit, or synchronous transit- guided-Quasi-Newton can be good approaches. • For intermediate systems: string method or nudged elastic band • For large systems with many DOF or to compute free energies: steered MD, metadynamics, umbrella sampling • Coordinate selection and initial guess for a reaction pathway are almost always challenging.
  • 44.
  • 45.
    MIT 10.637 Lecture 6 Transition statetheory Reaction coordinate Energy E a kBT Reactants Products Transition State Activation energy is one piece of the puzzle – full exploration of TST allows us to calculate absolute rates, through prefactor calculations. A potential energy surface connects the reactants and products: this is a QM concept! The activated complex or transition state is at the maximum potential energy. Once configuration is achieved, fast decay to products. Concentration of the transition state is dependent upon equilibrium with reactants, vibration along the path. 45
  • 46.
    MIT 10.637 Lecture 6 The Eyringequation Entropy of activation term Enthalpy of activation term Macroscopic Eyring equation: Microscopic Eyring equation: These are the molecular partition functions of A and B and the activated complex, AB. The Eyring equation is a rigorous method of obtaining kinetics from free energy of activation and may be derived from statistical mechanics (not here):
  • 47.
    MIT 10.637 Lecture 6 The Eyringequation and TST In the derivation of the Eyring equation (not shown), one vibrational frequency is stripped out of the molecular partition function for the activated complex. This is assumed to be the mode that carries the complex over the barrier to the products. Also: note that TST assumes once the activated complex is formed, it readily proceeds to products and does not cross back, which can be inaccurate.
  • 48.
    MIT 10.637 Lecture 6 Interpretation of 48 Thepotential energy surface is represented by changes in the internal coordinates of the molecule(s). These correspond to vibrational modes that alter internal coordinates or translation of the reactants with respect to each other. At the TS, we have one “molecule” and motion is characterized by the vibrational modes. One imaginary vibrational frequency characterizes the transition state and correlates to the attempt frequency in our derivation of the Eyring equation. Imaginary frequency of the transition state for HCNHNC isomerization
  • 49.
    MIT 10.637 Lecture 6 Partition functionreview # Maxwell-Boltzmann E Fraction molecules in state i. b= 1/kBT energy of the state Boltzmann distribution Molecular partition function sum over j states: sum over i levels: 1 2 3 4 5 6 1 2 3 4 5 6g=5 g=1
  • 50.
    MIT 10.637 Lecture 6 Molecular partitionfunction contributions are derived from solving eigenvalue problems in quantum mechanics.
  • 51.
    MIT 10.637 Lecture 6 Molecular degreesof freedom Linear Nonlinear Translational 3 3 Rotational 2 3 Vibrational 3N-5 3N-6 For zero field, potential energy derived only from vibrational coordinates. e.g. H2O: 3 normal modes symmetric stretch asymmetric stretchbend
  • 52.
    MIT 10.637 Lecture 6 Vibrational zero-pointenergy n=0 n=1 n=2 1 2 ~! 3 2 ~! 5 2 ~! From Q.M., molecules have zero point energy: ZPE Reaction coordinate + B P AB‡ A E0 Ea Energy Enthalpy of activation:
  • 53.
    MIT 10.637 Lecture 6 Contributions toprefactors DOF form f(TX) if if Order ½ ½ 0 1 1 10 108 cm-1 53
  • 54.
    MIT 10.637 Lecture 6 Prefactors fromquantum chemistry • Must have structures that are at minima or saddle points – vibrationally characterize them to get zero point energy and vibrational partition function. • ZPE corrects the enthalpy of activation. • Rotational partition function may also be calculated and its contribution is added to the partition functions. • Electronic terms are neglected. • Translational terms only depend on temperature.
  • 55.
    MIT 10.637 Lecture 6 Summary • Itis possible to use a saddle point structure and minimum energy structures to obtain absolute rates of reaction by calculating prefactors from statistical mechanics. • However, relative rates, where the prefactors are assumed to be the same for different reactions are typically preferred unless the method is extremely accurate.