Chapter 3 Calculation Methods

Transcription

1 Chapter 3 Calculation Methods Molecular Mechanics HyperChem uses two types of methods in calculations: molecular mechanics and quantum mechanics. The quantum mechanics methods implemented in HyperChem include semi-empirical, ab initio, and density functional quantum mechanics methods. The molecular mechanics and semi-empirical quantum mechanics methods have several advantages over ab initio and density functional methods. Most importantly, these methods are fast. While this may not be important for small molecules, it is certainly important for biomolecules. Another advantage is that for specific and well-parameterized molecular systems, these methods can calculate values that are closer to experiment than lower level ab initio and density functional techniques. The accuracy of a molecular mechanics or semi-empirical quantum mechanics method depends on the database used to parameterize the method. This is true for the type of molecules and the physical and chemical data in the database. Frequently, these methods give the best results for a limited class of molecules or phenomena. A disadvantage of these methods is that you must have parameters available before running a calculation. Developing parameters is time-consuming. The ab initio or density functional methods may overcome this problem. However they are slower than any molecular mechanics and semi-empirical methods. Molecular mechanical force fields use the equations of classical mechanics to describe the potential energy surfaces and physical properties of molecules. A molecule is described as a collection of atoms that interact with each other by simple analytical functions. This description is called a force field. One component of a force field is the energy arising from compression and stretching a bond. 21

2 This component is often approximated as a harmonic oscillator and can be calculated using Hooke s law. 1 V spring = --K (7) 2 r ( r r 0 ) 2 The bonding between two atoms is analogous to a spring connecting two masses. Using this analogy, equation 7 gives the potential energy of the system of masses, V spring, and the force constant of the spring, K r. The equilibrium and displaced distances of the atoms in a bond are r 0 and r. Both K r and r 0 are constants for a specific pair of atoms connected by a certain spring. K r and r 0 are force field parameters. The potential energy of a molecular system in a force field is the sum of individual components of the potential, such as bond, angle, and van der Waals potentials (equation 8). The energies of the individual bonding components (bonds, angles, and dihedrals) are functions of the deviation of a molecule from a hypothetical compound that has bonded interactions at minimum values. E Total = term 1 + term term n (8) The absolute energy of a molecule in molecular mechanics has no intrinsic physical meaning; E Total values are useful only for comparisons between molecules. Energies from single point calculations are related to the enthalpies of the molecules. However, they are not enthalpies because thermal motion and temperaturedependent contributions are absent from the energy terms (equation 8). Unlike quantum mechanics, molecular mechanics does not treat electrons explicitly. Molecular mechanics calculations cannot describe bond formation, bond breaking, or systems in which electronic delocalization or molecular orbital interactions play a major role in determining geometry or properties. This discussion focuses on the individual components of a typical molecular mechanics force field. It illustrates the mathematical functions used, why those functions are chosen, and the circumstances under which the functions become poor approximations. Part 2 of this book, Theory and Methods, includes details on the implementation of the MM+, AMBER, BIO+, and OPLS force fields in HyperChem. 22 Chapter 3

3 Bonds and Angles HyperChem uses harmonic functions to calculate potentials for bonds and bond angles (equation 9). V stretch = ( r r 0) 2 V bend = K θ ( θ θ 0 ) 2 K r bond angle Example: For the AMBER force field, a carbonyl C O bond has an equilibrium bond length of Å and a force constant of 570 kcal/mol Å 2. The potential for an aliphatic C C bond has a minimum at Å. The slope of the latter potential is less steep; a C C bond has a force constant of 310 kcal/mol Å 2. (9) K r = 570 kcal/mol Å 2 ; r 0 = 1.229Å K r = 310 kcal/mol Å 2 ; r 0 = 1.526Å bond length (Å) Calculation Methods 23

4 A Morse function best approximates a bond potential. One of the obvious differences between a Morse and harmonic potential is that only the Morse potential can describe a dissociating bond. Morse harmonic bond length (Å) The Morse function rises more steeply than the harmonic potential at short bonding distances. This difference can be important especially during molecular dynamics simulations, where thermal energy takes a molecule away from a potential minimum. In light of the differences between a Morse and a harmonic potential, why do force fields use the harmonic potential? First, the harmonic potential is faster to compute and easier to parameterize than the Morse function. The two functions are similar at the potential minimum, so they provide similar values for equilibrium structures. As computer resources expand and as simulations of thermal motion (See Molecular Dynamics, page 71) become more popular, the Morse function may be used more often. 24 Chapter 3

5 Torsions In molecular mechanics, the dihedral potential function is often implemented as a truncated Fourier series. This periodic function (equation 10) is appropriate for the torsional potential. V dihedrals = dihedrals V n 1+ cos( nφ φ 2 0 ) (10) In this representative dihedral potential, V n is the dihedral force constant, n is the periodicity of the Fourier term, φ 0 is the phase angle, and φ is the dihedral angle. Example: This example of an HN C(O) amide torsion uses the AMBER force field. The Fourier component with a periodicity of one (n = 1) also has a phase shift of 0 degrees. This component shows a maximum at a dihedral angle of 0 degrees and minima at both 180 and 180 degrees. The potential uses another Fourier component with a periodicity of two (n = 2). sum n=1 n=2 dihedral angle (degrees) The relative sizes of the potential barriers indicate that the V 2 force constant is larger than the V 1 constant. The phase shift is 180 degrees for the Fourier component with a two-fold barrier. Minima occur at 180, 0, and 180 degrees and maxima at 90 and 90 Calculation Methods 25

6 degrees. Adding the two Fourier terms results in potential with minima at 180, 0, and 180 degrees and maxima at 90 and 90 degrees. (The sum potential is shifted by 2 kcal/mol to make this illustration legible.) Note that the addition of V to V shows that 1 2 the cis conformation (dihedral hnco = 0 degrees) is destabilized relative to the trans conformation (dihedral hnco = 180 degrees). van der Waals Interactions and Hydrogen Bonding A 6 12 function (also known as a Lennard-Jones function) frequently simulates van der Waals interactions in force fields (equation 11). V VDR = i< j A ij 12 R ij B ij 6 R ij (11) R ij is the nonbonded distance between two atoms. A ij, and B ij are van der Waals parameters for the interacting pair of atoms. The R 6 term describes the attractive London dispersion interaction between two atoms, and the R 12 term describes the repulsive interaction caused by Pauli exclusion. The 6 12 function is not as appropriate physically as a 6-exponential function, but it is faster to compute. The AMBER force field replaces the van der Waals by a potential for pairs of atoms that can participate in hydrogen bonding (equation 12). The hydrogen bond potential does not contribute significantly to the hydrogen bonding attraction between two atoms; rather, it is implemented to fine-tune the distances between these atoms. V Hbonds = Hbonds C ij 12 R ij D ij 10 R ij (12) 26 Chapter 3

7 Example: In this example, the van der Waals (6 12) and hydrogen bond (10 12) potentials are quickly damped interatomic distance (Å) The attraction for two neutral atoms separated by more than four Ångstroms is approximately zero. The depth of the potential wells is minimal. For the AMBER force field, hydrogen bonds have well depths of about 0.5 kcal/mol; the magnitude of individual van der Waals well depths is usually less. Electrostatic Potential This is a typical function for electrostatic potential (equation 13). V EEL = q i q j εr i< j ij (13) In this model of electrostatic interactions, two atoms (i and j) have point charges q i and q j. The magnitude of the electrostatic energy (V EEL ) varies inversely with the distance between the atoms, R ij. The effective dielectric constant is ε. For in vacuo simulations or simulations with explicit water molecules, the denominator equals εr ij. In some force fields, a distance-dependent dielectric, where the denominator is εr ij R ij, represents solvent implicitly. Calculation Methods 27

8 Example: For nonbonded interaction between two atoms having point charges of and e, a distance-dependent dielectric, compared to a constant dielectric, reduces long range electrostatic interactions. Unlike van der Waals and hydrogen bond potentials, the magnitude of electrostatic potential energy between two atoms is large and remains significant over long nonbonded distances. distance-dependent dielectric constant dielectric interatomic distance (Å) United versus All Atom Force Fields Because of the restricted availability of computational resources, some force fields use United Atom types. This type of force field represents implicitly all hydrogens associated with a methyl, methylene, or methine group. The van der Waals parameters for united atom carbons reflect the increased size because of the implicit (included) hydrogens. United Atom force fields are used often for biological polymers. In these molecules, a reduced number of explicit hydrogens can have a notable effect on the speed of the calculation. Both the BIO+ and OPLS force fields are United Atom force fields. AMBER contains both a United and an All Atom force field. 28 Chapter 3

9 Caution: If you are new to computational chemistry, do not use United Atoms for AMBER calculations. This HyperChem option is available for researchers who want to alter atom types and parameters for this force field. Cutoffs Force field calculations often truncate the nonbonded potential energy of a molecular system at some finite distance. Truncation (nonbonded cutoff) saves computing resources. Also, periodic boxes and boundary conditions require it. However, this approximation is too crude for some calculations. For example, a molecular dynamic simulation with an abruptly truncated potential produces anomalous and nonphysical behavior. One symptom is that the solute (for example, a protein) cools and the solvent (water) heats rapidly. The temperatures of system components then slowly converge until the system appears to be in equilibrium, but it is not. A relatively simple method for alleviating some of the nonphysical behaviors caused by imposing a nonbonded cutoff is to use a potential switching function (equation 14). WhenR ij R on, E EEL = E EEL 1 WhenR on < R ij R off E ( R EEL E off R ij ) 2 ( R off + 2R ij 3R on ) = EEL ( ) 3 R off R on (14) WhenR ij > R off, E EEL = E EEL 0 These functions allow the nonbonded potential energy to turn off smoothly and systematically, removing artifacts caused by a truncated potential. With an appropriate switching function, the potential function is unaffected except in the region of the switch. Example: For two atoms having point charges of and e and a constant dielectric function, the energy curve shows a switching function turned on (R on ) at a nonbonded distance of 10 Å and off (R off ) at a distance of 14 Å. Compare the switched potential with the abruptly truncated potential. Calculation Methods 29

10 switched cutoff 10Å cutoff no cutoff interatomic distance (Å) HyperChem also provides a shifting potential for terminating nonbonded interactions (equation 15). 2 2R ij E EEL = E EEL R off 4 R ij R off (15) In an attempt to remedy truncation problems, a shifting potential multiplies the nonbonded electrostatic potential by a function that goes to zero. That is, the potential is shifted to zero at the cutoff R off. Unlike the switching function, the shifted potential does not apply to van der Waals interactions. 30 Chapter 3

11 Quantum Mechanics This section provides an overview and review of quantum mechanics calculations. The information can help you use Hyper- Chem to solve practical problems. For quantitative details of quantum mechanics calculations and how HyperChem implements them, see the second part of this book, Theory and Methods. Ab initio quantum mechanics methods have evolved for many decades. The speed and accuracy of ab initio calculations have been greatly improved by developing new algorithms and introducing better basis functions. Density functional methods are newer methods that have a lot in common with ab initio methods. They use density functional theory (DFT) to predict how the energy depends on the density rather than the wavefunction. In particular, they replace the Hartree- Fock potential of ab initio calculations with an exchange-correlation potential that is a functional of the electron density. They include the effects of electron correlation and hence can, in prinshifted cutoff constant dielectric interatomic distance (Å) Quantum Mechanics Caution: Comparing the shifted constant dielectric to a constant dielectric function without a cutoff shows that the shifted dielectric, unlike a switching function, perturbs the entire electrostatic energy curve, not only the region near the cutoff. Calculation Methods 31