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Other Popular Force Fields and MM Software. CHARMM, OPLS, and GROMOS

Other Popular Force Fields and MM Software. CHARMM, OPLS, and GROMOS

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Molecular Mechanics: Method and Applications







(-G(d) basis set) and on Monte Carlo simulations for hydrated ions (Jorgensen and TiradoRives ). The OPLS-AA (all atom) force field (Jorgensen et al. ) retain most of bond

stretch and angle bending parameters from AMBER all atom force fields, but torsion and nonbonding constants are reparameterized utilizing both experimental and ab initio (RHF/-G∗

level) data. This force field was improved for peptides by means of refitting the key Fourier torsional coefficients using accurate ab initio data (LMP/cc-pVTZ(-f)//HF/-G∗ ∗ level) as the

target (Kaminski et al. ).

Most popular program sets utilize various force fields, not only those developed in the

research groups of the authors of the program with the same title. The software suite GROMACS (Groningen MAchine for Chemical Simulation) was developed by the Berendsen group

at the University of Groningen, The Netherlands, starting in the early s. This fast program

for molecular dynamics simulation does not have a force field of its own, but is compatible with

the GROMOS, OPLS, and AMBER force fields. The GROMACS (Van der Spoel et al. )

was developed and optimized especially for use on PC-clusters. It originated from the Fortran

package GROMOS developed by van Gunsteren and Berendsen (Van Gunsteren et al. ).

The GROMOS force fields are united atom force fields, i.e., without explicit aliphatic (nonpolar) hydrogens. The latest version, GROMOS (Christen et al. ), utilizes new versions of

GROMOS force field sets (A and A) based on extensive reparameterization of the previous GROMOS force field (Oostenbrink et al. ). In contrast to the parameterization of most

of other biomolecular force fields, this parameterization of the GROMOS force field is based

primarily on reproducing the free enthalpies of hydration and apolar solvation for a range of

organic compounds.

Very popular for the computations of wide range of macromolecular systems are the

CHARMM (Chemistry at HARvard Macromolecular Mechanics) program and force field sets.

The computations include energy minimization, molecular dynamics, and Monte Carlo simulations. CHARMM originated with Martin Karplus’s group at Harvard University; the first

publication of the CHARMM program and force field was in  (Brooks et al. ). Several versions of both software and force field have since been released. Improvement to the

programs and force fields have progressed to date. The mathematical expressions for calculation of CHARMM energy are nearly the same as “minimalist” ones for other force fields

(MacKerell ). The minor differences with, e.g., AMBER or OPLS force fields relate to

inclusion of harmonic improper and Urey-Bradley terms. The parameters of various versions

of force field are consistently adjusted with emphasis to specific molecules to be applied.

The first versions of CHARMM force fields were aimed primary towards protein molecular dynamics simulations. Then, the special adjustments were performed for the simulations

of the nucleic acids (Foloppe and MacKerell ), and for lipids (Klauda et al. ).

The improvement of parameters for specific molecules has continued, while other parameters can be used from the previous versions. Raman et al. () recently extended the

CHARMM additive carbohydrate all-atom force field to enable modeling glycosidic linkages

in carbohydrates involving furanoses. As in most of popular modern force fields, both the

ab initio quantum mechanics computations and experimental data for solid and liquid states

are used in adjustment of the parameters of CHARMM force fields. The distinct feature of

this adjustment relates to charge adscription. Like OPLS force fields (Jorgensen et al. ),

CHARMM utilizes the so-called supramolecular approach. The partial atomic charges are

adjusted to reproduce ab initio (HF/-G∗ level) minimum interaction energies and geometries of model compounds with water or for model compound dimers (with energy and distance



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







Molecular Mechanics: Method and Applications



corrections for reproducing the correct experimental densities). On the other hand, the use

of ab initio data on small clusters to optimize the van der Waals parameters leads to poorly

condensed phase properties. This requires that the optimization of these parameters be performed by empirical fitting to reproduce thermodynamics properties from condensed phase

simulations, taking into account the relative values of the parameters obtained from highlevel ab initio data on interactions of the model compounds with rare gases (Klauda et al.

).

It is interesting to mention that multiple adjustments of different force fields mentioned

above resulted in an increase of similarity in both corresponding parameters values and

the results of applications to complex molecular systems (such as protein and nucleic acid

fragments), but many differences remain.



Conclusions

The MM computation approach to the study of the molecular systems arose about a half of century ago and today has become a useful and powerful research tool for many branches of natural

science. During this period of development, the number of publications utilizing this approach

increased by a few orders of magnitude. This increase continues up to present, followed by

the appearance of new journals and a flow of MM publications into the journals of a wide

area of coverage. The MM considers the molecular systems via a classic Newtonian mechanics

approach, i.e., using classical approximation to essentially quantum systems. The MM suggests

representation of the system energy as a sum of terms responsible for interactions of various

physical nature.

The minimalist MM approach involves the simple expressions for chemical bond stretching, bond angle bending, rotation about the bond, and nonbonded van der Waals and Coulomb

contributions calculated via additive atom-atom scheme. The force fields (sets of mathematical

formulae and numerical parameters) are derived from simple physical considerations followed

by adjustment to the experimental data and precise quantum mechanics results. The additivity of the terms responsible for separate atom contributions and for interactions of different

physical nature, and transferability of the parameters between atoms and molecules of similar structure are the main assumptions of the MM approach. Development of the approach

results in elaboration of a variety of computer software and force fields for simulating different

molecular systems of biological, medical, and industrial importance. Various force fields differ in the complexity of mathematical expressions and in the relative role of experimental and

quantum mechanics results used for the parameter adjustment. The development of algorithms

for MM applications (including those for molecular dynamics and Monte Carlo techniques) and

the increase of computer power has enabled researchers to approach such complex problems

as predicting the pathways of the formation of the biopolymer’s three-dimensional structure,

molecular recognition in various biological processes, and assistance in creation of compounds

with desirable properties (including drugs and industrial materials). Several force fields have

been elaborated recently, going beyond the atom-atom approach and the additivity principle.

There are programs utilizing the united atom approach to energy calculations and a hierarchical

procedure for consideration of the complex systems. It may be predicted that the role of MM

computations in the pure and applied science will arise synchronously with arising the role of

computers in the human life.



Molecular Mechanics: Method and Applications







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











Molecular Mechanics: Method and Applications



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



 Molecular Structure and

Vibrational Spectra

Jon Baker

Parallel Quantum Solutions, Fayetteville, Arkansas, USA



Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

He+

 : An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Newton–Raphson Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Hessian Matrix and Hessian Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Transition State Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Choice of Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Modified Newton–Raphson Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

GDIIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Geometry Optimization and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Performance for Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Minimization: An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Transition State Searches: An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Using Optimized Potential Scans in Transition State Searches . . . . . . . . . . . . . . . . . . . .

Comparison of Experimental and Theoretical Geometries . . . . . . . . . . . . . . . . . . . . . . .

Geometry Optimization of Molecular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Geometry Optimization in the Presence of External Forces . . . . . . . . . . . . . . . . . . . . . .





































Molecular Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,-Dichloroethane: An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Scaled Quantum Mechanical Force Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,-Dichloroethane: A Further Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Density Functional Theory and Weight Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . .















Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 



J. Leszczynski (ed.), Handbook of Computational Chemistry, DOI ./----_,

© Springer Science+Business Media B.V. 











Molecular Structure and Vibrational Spectra



Abstract: This chapter deals with two very important aspects of modern ab initio computational chemistry: the determination of molecular structure and the calculation, and

visualization, of vibrational spectra. It deals primarily with the practical aspects of determining molecular structure and vibrational spectra computationally. Both minima (i.e., stable

molecules) and transition states are discussed, as well as infrared (IR), Raman, and vibrational

circular dichroism (VCD) spectra, all of which can now be computed theoretically.



Introduction

This chapter deals with two very important aspects of modern ab initio computational chemistry: the determination of molecular structure and the calculation, and visualization, of vibrational spectra. The two things are intimately related as, once a molecular geometry has been

found (as a stationary point on a potential energy surface at whatever level of theory is being

used) it has to be characterized, which usually means that it has to be confirmed that the structure is a genuine minimum. This of course is done by vibrational analysis, i.e., by computing

the vibrational frequencies and checking that they are all real.

A large percentage of the total expenditure in CPU cycles devoted to computational chemistry (variously estimated at between % and %) is spent optimizing geometries. In order

to calculate various molecular properties, one first needs a reliable molecular structure so this

is perhaps not surprising. Algorithms for geometry optimization are now highly advanced and

usually very efficient and most of the quantum chemistry programs available for general use

have solid and reliable geometry optimization modules. They also nearly all have analytical second derivatives, at least for the most common theoretical methods, which makes it relatively

straightforward to compute vibrational frequencies once a structure has been found.

In this chapter I deal primarily with the practical aspects of determining molecular structure

and vibrational spectra computationally. I consider both minima (i.e., stable molecules) and

transition states, as well as infrared (IR), Raman, and vibrational circular dichroism (VCD)

spectra, all of which can now be computed theoretically. The program used to carry out the

calculations presented here is the PQS package developed by Parallel Quantum Solutions (PQS

), although any modern general purpose package (e.g. Gaussian, Turbomole, GAMESS)

would do just as well. As the name implies, all the major ab initio functionality of this package

is fully parallel, including energies, gradients, and second derivatives. PQS was chosen because

(a) it is the package I actually use in all my application work; and (b) I am one of its principal

authors. A review of the capabilities and parallel efficiency of the PQS package was published

recently (Baker et al. ).

I have elected to use a standard level of ab initio theory for all of the examples presented

in this chapter, namely, density functional theory (Hohenberg and Kohn ; Kohn and Sham

) (DFT) using the BLYP (Becke ) (see also Hertwig and Koch ()) functional and

the -G∗ (Ditchfield et al. ) basis set (BLYP/-G∗ ). DFT is now the method of choice

for routine chemical applications, and BLYP – despite the large number of functionals developed since – is still one of the most popular. Many of the techniques and pitfalls in locating

stable geometries are essentially independent of method, although DFT has its own issues as

a result of the numerical quadrature required to handle many of the integrals; this will be discussed in some detail later. DFT is so popular that Hartree–Fock theory, which used to be the

standard approach throughout the s, has now almost disappeared other than as a precursor



Molecular Structure and Vibrational Spectra







for higher level post-SCF calculations. At the time of writing, the latter are beginning to make

a bit of a comeback as the limitations of DFT are being reached, in particular its inability to

systematically improve the wavefunction (and hence the results). A good introduction to DFT

for those, like me, whose background is in traditional quantum chemistry is the now classic

 paper from (Johnson et al. ).

The Born–Oppenheimer approximation (Born and Oppenheimer ) (see also Wikipedia

()) is used throughout. This extremely important approximation, namely that electrons,

being so much less massive, respond instantaneously to the motion of the nuclei, underpins

the whole concept of the potential energy surface (PES). Stable arrangements of nuclei (stable or meta-stable molecules) correspond to local minima on the PES, first-order saddle points

connecting two different minima correspond to transition states, and the “valleys” joining transition states to minima correspond to “reaction paths.” Electronic excitation is represented by

a “jump” from one potential energy surface to another. We will be spending all of our time on

the lowest energy (ground-state) PES.



Molecular Structure

From a theoretical point of view the determination of a “molecular structure” means determining the geometry (i.e., the positions of atoms relative to one another) at a minimum (the bottom

of a well) on the ground-state potential energy surface. Although a prerequisite, locating a minimum does not automatically mean that you have found the structure of a stable molecule; this

depends on the barrier height, i.e., the energy required to get out of the well. The barrier height

can be determined by locating the transition state, which for our purposes is defined as the

highest energy point on the lowest energy path between reactants and products. (In a barrierless reaction, this will effectively be the energy of the products, assuming the reactant is taken

to have the lower energy.) As a rough rule of thumb, if the barrier height is less than about

– kcal mol− , the system is kinetically unstable at room temperature.

Determining the geometry of a stable molecule therefore requires knowledge of both the

minimum itself and the barrier height (i.e., effectively locating the transition state) for all possible decomposition reactions. Of course, the structures of many stable molecules can be derived

from “chemical intuition” based on preexisting knowledge, but for new structures in particular

the transition state is equally important and the practicing theoretician must be able to find

both. In this section we discuss the methods available to do so.



He+

: An Illustrative Example



The simplest possible molecular orbital approach to the bonding in the hydrogen molecule (H )

would consider the overlap between two s atomic orbitals on each hydrogen producing two

molecular orbitals (MOs), one of which is bonding (the in-phase combination) and the other

antibonding (the out-of-phase combination). H is stable because its two electrons occupy the

bonding orbital, giving an overall energy lowering compared to two separate H atoms.

Similar considerations applied to helium would suggest, quite rightly, that He would not be

stable, since both the bonding and antibonding MOs would be occupied, giving no net bonding.

(In fact, as is well known, the antibonding MO is more antibonding than the bonding MO is







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