Introduction to Geometry Optimization Computational Chemistry lab 9
Determination of the molecule configuration H H Diatomic molecule determine the interatomic distance H O H Triatomic molecule determine distances and the angle H N H H Four-atomic molecule determine angles and distances Optimal configurations correspond to minima of the energy as a function of 3N- 6 parameters for an N-atomic molecule (3N atomic coordinates-3 c.o.m coordinates - 3 angles)
Solution of the Schrödinger equation in the Born-Oppenheimer approimation: H tot = (T n + V n ) + T e + V ne + V e = (H n ) + H e lectronic S: H e Ψ (r,r)= e (R) Ψ (r,r) Ab-initio or semi-empirical solution e (R) electronic energy R nuclear coordinates V n (R) niclei-nuclei interaction (R)=V n (R)+ e (R) = potential energy surface (PS) Molecular mechanics: PS molecular force field
Potential nergy Surface (PS) A hypersurface in 3N-6 -dimensional hyperspace, where N is the number of atoms. Minima equilibrium structures of one or several molecules Saddle points - transition structures connecting equilibrium structures. Global Minimum - that point that is the lowest value in the PS
One configuration is nown for H One configuration is nown for H O The planar and pyramidal configurations are nown for NH 3 The number of local minima typically goes eponentially with the number of variables (degrees of freedom). Combinatorial plosion Problem Possible Conformations (3 n ) for linear alanes CH 3 (CH ) n+ CH 3 n = 3 n = 9 n = 5 43 n = 59,49 n = 5 4,348,97
Global minimum - Mariana Trench in the Pacific Ocean
Optimization/Goals Find the Local Minimum Structure Find the Global Minimum Structure Find the Transition State Structure We would lie to Find the Reaction Pathway Connecting Two Minima and Passing through the Transition State Structure Geometry optimization is the name for the procedure that attempts to find the configuration of minimum energy of the molecule.
One-dimensional optimization f 3 maimum f f ()= stationary point f ()< maimum minimum 5 f ()> minimum f' - f'' -5 -.5..5..5..5
Multidimensional Optimization Force: F=- (Potential energy) - gradient The coordinates can be Cartesian (,y,z..), or internal, such as bondlength and angle displacements) f= - Stationary Point (minimum, maimum, or saddle point)
Hessian matri A matri of second-order derivatives of the energy with respect to atomic coordinates (e.g., Cartesian or internal coordinates) Sometimes called force matri matri size of (3N-6)(3N-6) ( H ij ( f ) f i j = Quadratic approimation Approimate the comple energy landscape by harmonic potentials around a stationary point (,... () () () () n [ (,... n ) = ] n n () () = + () (),... n) (,... n ) Hij( )( i i )( j j ) i= j= )
Hessian matri diagonalization (eigenproblem) igenvalues igenvectors All ε > ( ω real) minimum ω ε l ( ) j ε = mω - vibrational frequencies n ( ) ( l j give normal coordinates q = li () () (,... ) = (,... + ε q n n ) n = All ε < ( ω imaginary) maimum i= = ) n i= ( i H ij l ( ) i () i ε <> ( ω imaginary/real) saddle point )
Steepest Descent Steepest descent direction g=- Ε g - 3N-dimensional vector = + λ g / g i+ i i i i Advantage: fast minimization far from the minimum Disadvantage: slow descent along narrow valleys slow convergence near minima In HYPRCHM if ( i + ) < ( i) then i+ =.λi else λi+ = History gradient information is not ept λ. 5λ i
Conjugate Gradient Methods Search direction is chosen using history gradient information Initial direction the steepest descent direction, h = g =- Ε = + ξh i+ i i ξ is defined by a one-dimensional minimization along the search direction. hi+ = gi+ + γ i+ hi g i Fletcher-Reeves method: γ + i+ = gi ( i+ gi ) g Pola-Ribiere method: γ i+ = g g i+ Pola-Ribiere may be superior for nonquadratic functions. i steepest conjugate Gradient history is equivalent to implicit use of the Hessian matri.
plicit use of Hessian matri. Quadratic approimation : ( ) = q ε - normal coordinates, and l Newton-Raphson Methods - ( = eigenvalues and eigenvectors of = n q l ), = g n ( F q = n = F l ε q the Hessian matri at, ), One-step optimization of quadratic functions q = F / ε For arbitrary functions n i+ = i + l ( i) F ( i ) / ε ( i = Steeprst descent direction for ε ( i ) > ) Newton- Raphson conjugate Finds the closest stationary point (either minimum, maimum, or saddle point).
Numerical calculation (3N) calculations of energy Diagonalization - (3N) 3 operations Memory - (3N) tremely epensive for large molecules. Obtaining, storing, and diagonalaizing the Hessian Bloc Diagonal Newton-Raphson z z y z z y y y z y z z y z z y y y z y Calculation - 9N Diagonalization - 7N Memory 9N
ε Gaussian The Hessian matri is calculated and processed on the first step only It is updated using the computed energies and gradients Rational function optimization (RFO) step i+ = and l Berny algorithm i + n = l ( i F ( i ) ) ε ( ) λ λ<min(ε ) (λ< for ε <) - step always toward to a minimum. A linear search between the latest point and the best previous point. i eigenvalues and eigenvectors of the Hessian matri at RFO Newton-Raphson
Convergence criteria Gradient or force F=-g=- Ε Predicted displacement i+ - i Maimum force: ma(f)<threshold Root-mean-square (RMS) force: F <threshold Maimum displacement: ma( i+ - i ) <threshold RMS displacement: i+ - i <threshold
One configuration is nown for H One configuration is nown for H O The planar and pyramidal configurations are nown for NH 3 The number of local minima typically goes eponentially with the number of variables (degrees of freedom). Combinatorial plosion Problem Possible Conformations (3 n ) for linear alanes CH 3 (CH ) n+ CH 3 n = 3 n = 9 n = 5 43 n = 59,49 n = 5 4,348,97
Global minimization "building up" the structure: combining a large molecule from preoptimized fragments (protein from preoptimized aminoacids) conformational sampling: tae various starting points for local minimization (simulation of nature) None of these are guaranteed to find the global minimum!
Global Minimum- conformational sampling Molecular Dynamics Monte Carlo Simulated Annealing Tabu search Genetic Algorithms Ant Colony Optimizations Diffusion Methods Distance Geometry Methods
Heating above barriers System overcomes barriers at sufficiently high temperature T quilibrium (Boltzmann) distribution P(R)~ep(-(R)/( B T )) Higher probability to find the system near the global minimum
Molecular Dynamics Solve equations of motion d T v j = j( R) + ( ) dt m j τ T R j atomic coordinates d v j = R j atomic velocities dt 3 T = v j temperature B j m j T bath temperature τ bath relaation time The trajectory fills available phase space Monte Carlo. One atom is randomly "iced". The energy change = new - old is calculated 3. If < the new configuration is accepted 4. If > the new configuration is accepted with probability ep(- /( B T )) 5. Net atom is randomly "iced" Both methods lead to equilibrium averaged distribution in the limits of infinite time or number of steps
Simulated annealing If a slow cooling is applied to a liquid, this liquid freezes naturally to a state of minimum energy. bath temperature heating equilibration cooling ither Molecular-Dynamics or Monte-Carlo simulation can be used. Is guaranteed to find the global minimum if the equilibration is infinitely long and the cooling is infinitely slow.