Example questions for Molecular modelling (Level 4) Dr. Adrian Mulholland 1) Question. Two methods which are widely used for the optimization of molecular geometies are the Steepest descents and Newton-Raphson techniques. Without giving detailed mathematical descriptions, briefly outline the advantages and disadvantages of these two techniques. Outline Answer: The steepest descents method of energy minimization is a simple and effective method of improving the geometry of a molecular model (i.e. finding a lower energy structure). It moves straight downhill on the energy surface, taking steps in the direction of the negative of the gradient. This involves calculating the forces (gradients) on the atoms - the first derivative of the energy). The gradient is a vector (g), with one component for each coordinate. The step, s, is in the direction g. The steepest descents step is determined mostly by the largest forces. It is therefore good for relieving strain in a built model. Only first derivatives are required - this is an advantage for large molecules. Steepest descents provides fast minimization (big steps) when far from minimum, but small steps (slow minimization) when close to minimum. It has slow convergence (i.e. it will find the minimum energy structure only very slowly). It may also oscillate around a minimum and never actually reach the minimum. Because it only moves downhill on the energy surface, it is no use for TS searches. The Newton-Raphson method uses in addition second derivatives of the energy and can be a better minimization method. It is efficient for small molecules, and converges quickly, in contrast to steepest descents minimization. However, it makes the approximation that the energy surface has a quadratic form, which is often poor approximation for molecular potential energy surfaces, particularly far from minimum The calculation, inversion and storage of the Hessian matrix (i.e. the matrix of second derivatives) is computationally difficult for large molecules. It is possible to locate stationary points other than minima on potential energy surfaces with the Newton- Raphson method, and this can be useful for transition state optimization. The Newton-Raphson method works well if the initial geometry is close to a stationary point, but may not work well if it is not. Also, technically the method may be difficult to use for some types of problem. Technical difficulties arise because of the need to calculate the Hessian, which is a potentially very large matrix (no. of variables squared). The calculation of the second derivatives can be demanding (requiring a lot of computer time), and their storage can be difficult (a lot of computer memory is required). The Hessian must be inverted (diagonalized), which is a technical problem for more than 1000 variables (1000 1000 matrix), i.e. approx. 300 atoms. It is possible to avoid calculating the Hessian matrix at every step by either using a guessed matrix (e.g. the identity matrix) or by using a Hessian update scheme (constructs an approximate Hessian based on e.g. the gradient and coordinates at current and new points. Calculating the Hessian at a lower level of theory can also reduce the demands of the calculation. When dealing with large numbers of atoms, though, the Newton-Raphson method is not suitable because of the large computational requirements. A first-order method has to be used instead (first-order methods use first derivatives of the energy; second-order methods use first and second derivatives). 2) Question: Give reasons why calculation and analysis of the harmonic vibrational frequencies of molecules may be useful. Outline the process involved in calculation of harmonic vibrational frequencies.
There are several reasons for calculating vibrational frequencies of molecules, including: To identify the nature of a stationary point found by geometry optimization (e.g. a minimum or a saddlepoint). At a minimum, all the 3N-6 eigenvalues of the Hessian are positive, so all the calculated frequencies will be positive. To calculate the infrared and Raman spectra of a molecule To calculate thermodynamic properties of a molecule Vibrational frequencies are usually calculated by finding the normal modes of the molecule. 3) Question: Give an expression for a molecular mechanics potential energy function of the type used for standard macromolecular simulations. Include intra- and intermolecular terms. State clearly what each energy term represents, and its physical origin. How, typically, would the parameters of such a potential energy function be found? Discuss the range of applicability of a potential energy function of this type, its strengths and its limitations. Give brief examples of applications. What improvements could be made to a potential energy function of this type? Answer: The answer should discuss the harmonic function used to model bond stretching, the limitations of the harmonic potential (e.g. no dissociation; poor behaviour away from minimum); the harmonic valence angle bending; the cosine series for torsion angles, mentioning in each case the meaning of the symbols (i.e. force constants, barrier heights, phase factors), and the need for improper torsion terms. The final terms should be identified as the non-bonded interactions in the potential function, van der Waals and electrostatic terms. The answer should discuss which non-bonded interactions would be calculated (i.e. 1-4 interactions etc). In discussing the van der Waals terms, the repulsive and attractive interactions should be identified, and the cause (dispersion interactions and exchange repulsion) should be outlined. Atomic partial charges should be described and the ability of these models to reproduce hydrogen bond structures. Limitations of molecular mechanics force fields such as incomplete and potentially incorrect description of electrostatic interactions, lack of electronic polarizability (fixed atomic charges), inability to model reactions (incorrect functional form), dependence on parameterization (e.g. limited dataset), should be discussed. Possible improvements include: inclusion of atomic polarizability, different forms for bonding interactions; and combined quantum mechanics/molecular mechanics method (outline briefly) the latter allow modelling of reactions. 4) Question i. Write down the Lennard-Jones potential energy function typically used to model van der Waals interactions in molecular modelling of proteins, in terms of the collision diameter,, well-depth,, and interatomic separation, r.
ii. State briefly the meaning of and. iii. Sketch the form of this energy equation, labelling the axes as energy and interatomic separation. Indicate and on the sketch. Outline answer: i. V r ii. σ 4ε r 12 σ r This function contains two parameters: the well depth, (the depth of the energy minimum for the interaction), and the collision diameter, (the separation at which the interaction energy is zero). 6 iii. Sketch the curve: 5) What types of points on a potential energy surface are particularly relevant in understanding a chemical reaction? Transition state structures are saddlepoints on the potential energy surface; stable structures are minima on the potential energy surface. 6) Give reasons why calculation and analysis of the harmonic vibrational frequencies of molecules may be useful. Outline the process involved in calculation of harmonic vibrational frequencies. Identify the type of stationary point found by geometry optimization (e.g. is it a minimum or a saddlepoint?) IR and Raman spectra of a molecule Calculate zero-point energy important for some reactions Thermodynamic properties (e.g. vibrational energy contribution to energy barrier) Contribution of quantum tunnelling to a reaction barrier
7) Describe how you could confirm that a structure is a transition state structure. Also outline how you could show that the transition state really is the right one for the reaction of interest. Outline answer: Transition state structures are saddlepoints They can be transition states for chemical changes, or for conformational changes For a transition state structure (saddlepoint), one of the vibrational eigenvalues will be negative - it is a maximum in one direction In all other directions, it is a minimum The frequency is given by the square root of the eigenvalue, so formally there will be one imaginary frequency If there is one, and only one, imaginary frequency, the structure really is a transition state The atomic displacements for the imaginary frequency are very important - this normal mode corresponds to motion along the reaction coordinate For a TS structure, the atomic displacements for the imaginary frequency show which atoms are involved are involved in motion along the reaction coordinate A more detailed check is to calculate the steepest descent path from the TS: it should lead (in one direction) to the reactants and (and in the other direction) to the products. This path (calculated in mass-weighted coordinates from the TS) is called the intrinsic reaction coordinate 8) Briefly describe the following methods for calculating molecular energies and geometries (outline their advantages and disadvantages): (i) ab initio molecular orbital methods (ii) semiempirical molecular orbital methods (iii) molecular mechanics methods (iv) combined quantum mechanics/molecular mechanics methods Outline answer: Describe each type of method briefly. The answer should mention e.g. that semiempirical methods are more approximate than ab initio methods; both are quantum mechanical electronic structure methods; and that both can be applied to model chemical reactions unlike molecular mechanics. QM/MM methods combine QM and approaches and thus allow modelling of reactions in large systems such as enzymes. 9) Discuss the applicability and accuracy of the following methods to calculation of barriers for (a) conformational changes and (b) chemical reactions: molecular mechanics, Hartree-Fock ab initio molecular orbital calculations, semiempirical QM calculations; MP2 ab initio calculations. (e.g. for each method, discuss whether it is likely to give calculated rate constants close to experiment for conformational changes, and (separately) for chemical reactions. Outline answer: Molecular mechanics methods can calculate conformational properties, including energy barriers, quite well if suitably parameterized. They are not useful for modelling chemical reactions: for example, the use of a harmonic term for bonds means that they cannot break. Hartree-Fock ab initio molecular orbital methods can deal well with conformational barriers. This is because correlation energy does not change much in a conformational change because chemical bonding does not change. In a chemical reaction, on the other hand, the correlation energy changes significantly, so Hartree-Fock calculations are not very useful for chemical reactions: they tend to give barriers which are much too high.
Semiempirical molecular orbital methods can be applied to model conformational or chemical changes, but are more approximate than ab initio methods and so are prone to large errors. MP2 ab initio methods improve on Hartree-Fock by including electron correlation effects. Rate constants (for chemical or conformational changes) can be calculated using the Eyring equation. MP2 calculations would be expected to give rate constants close to experimental values. The other electronic structure (QM) methods would be likely to give results in poorer agreement with experiment. 10) Outline the essential features the TIP3P model used for simulations of liquid water. Discuss its strengths and weaknesses. Outline the features of more sophisticated water models for molecular simulations. Outline answer: TIP3P is a standard model of water for atomistic simulations. It is an invariant point charge model. Each atom has a fixed charge (q(o) = 0.834, (H)=0.417)). There are Lennard-Jones parameters for the oxygen only. Fixed geometry (based on experimental structure, O-H distance =0.09572 nm, angle H-O- Hw = 104.52º. Dipole moment of TIP3P (2.35D) is higher than the experimental result for a water molecule in the gas phase (1.85D). Closer to effective dipole of liquid water (2.6D). Interaction energy for TIP3P water dimer is 6.5 kcal/mol, higher than experiment ( 5.4 ± 0.7 kcal/mol). Liquid water is polarized, each molecule by its neighbours. TIP3P was developed by fitting to experimental properties of liquid water. It is a poor model for small numbers of water molecules, but reasonable for modelling liquid water. Better models can include more point charges to give a better description of the electrostatic interactions of water (e.g. TIP4P, TIP5P). More sophisticated still are models that include electronic polarization explicitly, i.e. polarizable water models. 10) Classical molecular dynamics simulations are an important technique in molecular modelling (e.g. of proteins). Describe the essential features of such molecular dynamics simulations. Outline answer: Classical MD determines atomic positions through solving Newton's equations of motion. Outline integration algorithms for MD, typical timesteps, use of SHAKE algorithm for increasing timestep. A molecular dynamics simulation requires the definition of a potential function, or a description of the terms by which the particles in the simulation will interact. This is usually referred to as a force field. Potentials may be defined at many levels of physical accuracy; those most commonly used in chemistry are based on molecular mechanics and embody a classical treatment of particle-particle interactions that can reproduce structural and conformational changes but usually cannot treat chemical reactions. The reduction from a fully quantum description to a classical potential entails two main approximations. The first one is the Born-Oppenheimer approximation, which states that the dynamics of electrons is so fast that they can be considered to react instantaneously to the motion of their nuclei. As a consequence, they may be treated separately. The second one treats the nuclei, which are much heavier than electrons, as point particles that follow classical Newtonian dynamics. In classical molecular dynamics the effect of the electrons is approximated as a single potential energy surface, usually representing the ground state. When treatment of electronic properties is required, potentials based on quantum mechanics are used; some techniques attempt to create hybrid classical/quantum ( QM/MM ) potentials where the bulk of the system is treated classically but a small region is treated as a quantum system, e.g. for modelling a chemical transformation.
11) Hartree-Fock ab initio calculations give an energy of 79.229 Hartree for ethane in its staggered conformation, and 79.224 Hartree for the eclipsed form. From these results, calculate the barrier to rotation in ethane in kj/mol, given that 1 H is equivalent to 2625.5 kj/mol. Comment on the fact that the barrier obtained by a Hartree-Fock calculation is found to be close to the experimental result. Outline answer: Energy barrier = 0.005 H = 13.1 kj/mol. A Hartree-Fock calculation gives a good answer in this case because the electron correlation energy does not change significantly during rotation around the C-C bond. Hartree-Fock calculations do not include the effects of electron correlation, and so overestimate electron-electron repulsion. Therefore the absolute energy calculated by a Hartree-Fock method is too high. However, for processes of conformational change, the correlation energy remains roughly constant, so HF calculations can give a useful prediction of relative energies. This is not the case for processes involving breaking or making of bonds.