Young D.C. - Computational chemistry (2001)(en)
.pdf172 19 REACTION RATES
Radiationless transition calculations are described in
M. Klessinger, Theoretical Organic Chemistry C. PaÂrkaÂni Ed., 581, Elsevier (1998).
J. Simons, J. Nichols, Quantum Mechanics in Chemistry 314, Oxford, New York (1997). V. V. Kocharovsky, V. V. Kocharovsky, S. Tasaki, Adv. Chem. Phys. 99, 333 (1997). Adv. Chem. Phys. P. Gaspard, I. Burghardt, Eds., 101 (1997).
R. H. Landau, Quantum Mechanics II; A Second Course in Quantum Theory Second Edition 314, John Wiley & Sons, New York (1996).
F. Bernardi, M. Olivucci, M. A. Robb, Chem. Soc. Rev. 25, 321 (1996).
L.Serrano-Andres, M. Merchan, I. Nebot-Gil, R. Lindh, B. O. Roos, J. Chem. Phys. 98, 3151 (1993).
Adv. Chem. Phys. M. Baer, C. ±Y. Ng, Eds., 82 (1992).
M. Baer, Theory of Chemical Reaction Dynamics Vol. II M. Baer Ed., 219, CRC (1985). A. A. Ovchinnikov, M. Y. Ovchinnokova, Adv. Quantum Chem. 16, 161 (1982).
C.Cohen-Tannoudji, B. Diu, F. LaloeÈ, Quantum Mechanics 1299, John Wiley & Sons, New York (1977).
E.E. Nikitin, Adv. Quantum Chem. 5, 135 (1970).
Statistical calculations are reviewed in
J. Troe, Adv. Chem. Phys. 101, 819 (1997).
D. A. McQuarrie, Adv. Chem. Phys. 15, 149 (1969).
Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems. David C. Young Copyright ( 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-33368-9 (Hardback); 0-471-22065-5 (Electronic)
20 Potential Energy Surfaces
In the chapter on reaction rates, it was pointed out that the perfect description of a reaction would be a statistical average of all possible paths rather than just the minimum energy path. Furthermore, femtosecond spectroscopy experiments show that molecules vibrate in many di¨erent directions until an energetically accessible reaction path is found. In order to examine these ideas computationally, the entire potential energy surface (PES) or an approximation to it must be computed. A PES is either a table of data or an analytic function, which gives the energy for any location of the nuclei comprising a chemical system.
20.1PROPERTIES OF POTENTIAL ENERGY SURFACES
Once a PES has been computed, it can be analyzed to determine quite a bit of information about the chemical system. The PES is the most complete description of all the conformers, isomers, and energetically accessible motions of a system. Minima on this surface correspond to optimized geometries. The lowest-energy minimum is called the global minimum. There can be many local minima, such as higher-energy conformers or isomers. The transition structure between the reactants and products of a reaction is a saddle point on this surface. A PES can be used to ®nd both saddle points and reaction coordinates. Figure 20.1 illustrates these topological features. One of the most common reasons for doing a PES computation is to subsequently study reaction dynamics as described in Chapter 19. The vibrational properties of the molecule can also be obtained from the PES.
In describing PES, the terms adiabatic and diabatic are used. In the older literature, these terms are used in confusing and sometimes con¯icting ways. For the purposes of this discussion, we will follow the conventions described by Sidis; they are both succinct and re¯ective of the most common usage. The term adiabatic originated in thermodynamics, where it means no heat transfer …dq ˆ 0†. An adiabatic PES is one in which a particular electronic state is followed; thus no transfer of electrons between electronic states occurs. Only in a few rare cases is this distinction noted by using the term electronically adiabatic. A diabatic surface is the lowest-energy state available for each set of nuclear positions, regardless of whether it is necessary to switch from one electronic state to another. These are illustrated in Figure 20.2.
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174 20 POTENTIAL ENERGY SURFACES
PES
IRC path
saddle point
local minimum
global minimum
FIGURE 20.1 Points on a potential energy surface.
The mathematical de®nition of the Born±Oppenheimer approximation implies following adiabatic surfaces. However, software algorithms using this approximation do not necessarily do so. The approximation does not re¯ect physical reality when the molecule undergoes nonradiative transitions or two
adiabatic paths
diabatic path
FIGURE 20.2 Adiabatic paths for bond dissociation in two di¨erent electronic states and the diabatic path.
20.2 COMPUTING POTENTIAL ENERGY SURFACES 175
electronic states are intimately involved in a process, such as Jahn±Teller distortions or vibronic coupling e¨ects.
Depending on the information desired, the researcher might wish to have a calculation follow either an adiabatic path or a diabatic path, as shown in Figure 20.2. Which potential energy surface is followed depends on both the construction of the wave function and the means by which wave function symmetry is used by the software. A single-determinant wave function in which wave function symmetry is imposed will follow the adiabatic surface where two states of di¨erent symmetry cross. When two states of the same symmetry cross, a single-determinant wave function will exhibit an avoided crossing, as shown in Figure 17.2. A multiple-determinant wave function, in which both states are in the con®guration space, will have the ¯exibility to follow the diabatic path. It is sometimes possible to get a multiple-determinant calculation (particularly MCSCF) to follow the adiabatic path by moving in small steps and using the optimized wave function from the previous point as the initial guess for each successive step.
20.2COMPUTING POTENTIAL ENERGY SURFACES
Computing a complete PES for a molecule with N atoms requires computing energies for geometries on a grid of points in 3N-6 dimensional space. This is extremely CPU-intensive because it requires computing a set of X points in each dimension, resulting in X 3N-6 single point computations. Because of this, PES's are typically only computed for systems with a fairly small number of atoms.
Some software packages have an automated procedure for computing all the points on a PES, but a number of technical problems commonly arise. Some programs have a function that halts the execution when two nuclei are too close together; this function must be disabled. SCF procedures often exhibit convergence problems far from equilibrium. Methods for ®xing SCF convergence problems are discussed in Chapter 22. At di¨erent points on the PES, the molecule may have di¨erent symmetries. This often results in errors when using software packages that use molecular symmetry to reduce computation time. The use of symmetry by a program can often be turned o¨. Computation of the PES for electronic excited states can lead to additional technical di½culties as described in Chapter 25.
The level of theory necessary for computing PES's depends on how those results are to be used. Molecular mechanics calculations are often used for examining possible conformers of a molecule. Semiempiricial calculations can give a qualitative picture of a reaction surface. Ab initio methods must often be used for quantitatively correct reaction surfaces. Note that size consistent methods must be used for the most accurate results. The speci®c recommendations given in Chapter 18 are equally applicable to PES calculations.
176 20 POTENTIAL ENERGY SURFACES
20.3FITTING PES RESULTS TO ANALYTIC EQUATIONS
Once a PES has been computed, it is often ®tted to an analytic function. This is done because there are many ways to analyze analytic functions that require much less computation time than working directly with ab initio calculations. For example, the reaction can be modeled as a molecular dynamics simulation showing the vibrational motion and reaction trajectories as described in Chapter 19. Another technique is to ®t ab initio results to a semiempirical model designed for the purpose of describing PES's.
Of course, the analytic surface must be fairly close to the shape of the true potential in order to obtain physically relevant results. The criteria on ®tting PES results to analytic equations have been broken down into a list of 10 speci®c items, all of which have been discussed by a number of authors. Below is the list as given by Schatz:
1.The analytic function should accurately characterize the asymptotic reactant and product molecules.
2.It should have the correct symmetry properties of the system.
3.It should represent the true potential accurately in the interaction regions for which experimental or nonempirical theoretical data are available.
4.It should behave in a physically reasonable manner in those parts of the interaction regions for which no experimental or theoretical data are available.
5.It should smoothly connect the asymptotic and interaction regions in a physically reasonable way.
6.The interpolating function and its derivatives should have as simple an algebraic form as possible consistent with the desired goodness of ®t.
7.It should require as small a number of data points as possible to achieve an accurate ®t.
8.It should converge to the true surface as more data become available.
9.It should indicate where it is most meaningful to compute the data points.
10.It should have a minimal amount of ``ad hoc'' or ``patched up'' character.
Criteria 1 through 5 must be obeyed in order to obtain reasonable results in subsequent calculations using the function. Criteria 6 through 10 are desirable for practical reasons. Finding a function that meets these criteria requires skill and experience, and no small amount of patience.
The analytic PES function is usually a summation of twoand three-body terms. Spline functions have also been used. Three-body terms are often polynomials. Some of the two-body terms used are Morse functions, Rydberg
BIBLIOGRAPHY 177
functions, Taylor series expansions, the Simons±Parr±Finlan (SPF) expansion, the Dunham expansion, and a number of trigonometric functions. Ermler & Hsieh, Hirst, Isaacson, and Schatz have all discussed the merits of these functions. Their work is referenced in the bibliography at the end of this chapter.
20.4FITTING PES RESULTS TO SEMIEMPIRICAL MODELS
The most commonly used semiempirical for describing PES's is the diatomics- in-molecules (DIM) method. This method uses a Hamiltonian with parameters for describing atomic and diatomic fragments within a molecule. The functional form, which is covered in detail by Tully, allows it to be parameterized from either ab initio calculations or spectroscopic results. The parameters must be ®tted carefully in order for the method to give a reasonable description of the entire PES. Most cases where DIM yielded completely unreasonable results can be attributed to a poor ®tting of parameters. Other semiempirical methods for describing the PES, which are discussed in the reviews below, are LEPS, hyperbolic map functions, the method of Agmon and Levine, and the mole- cules-in-molecules (MIM) method.
BIBLIOGRAPHY
Introductory discussions can be found in
J.B. Foresman, á. Frisch, Exploring Chemistry with Electronic Structure Methods Second Edition Gaussian, Pittsburgh (1996).
A.R. Leach Molecular Modelling Principles and Applications Longman, Essex (1996).
I.N. Levine, Physical Chemistry Fourth Edition McGraw Hill, New York (1995).
P.W. Atkins, Quanta Oxford, Oxford (1991).
T.Clark, A Handbook of Computational Chemistry John Wiley & Sons, New York (1985).
Books on potential energy surfaces are
V.I. Minkin, B. Y. Simkin, R. M. Minyaev, Quantum Chemistry of Organic Compounds; Mechanisms of Reactions Springer-Verlag, Berlin (1990).
P. G. Mezey, Potential Energy Hypersurfaces Elsevier, Amsterdam (1987).
J.N. Murrell, S. Carter, S. C. Farantos, P. Huxley, A. J. C. Varandas, Molecular Potential Energy Functions John Wiley & Sons, New York (1984).
Potential Energy Surfaces and Dynamics Calculations D. G. Truhlar, Ed., Plunum, New York (1981).
Review articles are
M. L. McKee, M. Page, Rev. Comput. Chem. 4, 35 (1993).
Adv. Chem. Phys. M. Baer, C.-Y. Ng, Eds., 82 (1992).
178 20 POTENTIAL ENERGY SURFACES
L.B. Harding, Advances in Molecular Electronic Structure Theory 1, 45, T. H. Dunning, Jr., Ed., JAI, Greenwich (1990).
K. Balasubramanian, Chem. Rev. 90, 93 (1990).
Advances in Molecular Electronic Structure Theory, Calculation and Characterization of Molecular Potential Energy Surfaces T. H. Dunning, Jr., Ed., JAI, Greenwich (1990).
New Theoretical Concepts for Understanding Organic Reactions J. BertraÂn, I. G. Csizmadia, Eds., Kluwer, Dordrecht (1988).
D. G. Truhlar, R. Steckler, M. S. Gordon, Chem. Rev. 87, 217 (1987). P. J. Bruna, S. D. Peyerimho¨, Adv. Chem. Phys. 67, 1 (1987).
T.H. Dunning, Jr., L. B. Harding, Theory of Chemical Reaction Dynamics Vol 1
M. Baer Ed. 1, CRC (1985).
P. J. Kuntz, Theory of Chemical Reaction Dynamics Vol 1 M. Baer Ed. 71, CRC (1985). D. M. Hirst, Adv. Chem. Phys. 50, 517 (1982).
Adv. Chem. Phys. K. P. Lawley, Ed., 42 (1980).
J.N. Murrell, Gas Kinetics and Energy Transfer, Volume 3 P. G. Ashmore, R. J. Donovan (Eds.) 200, Chemical Society, London (1978).
W. A. Lester, Jr., Adv. Quantum Chem. 9, 199 (1975).
G. G. Balint-Kurti, Adv. Chem. Phys. 30, 137 (1975).
Reactions involving crossing between excited state surfaces are discussed in
M. Klessinger, Theoretical Organic Chemistry C. PaÂrkaÂni Ed., 581, Elsevier (1998). F. Bernardi, M. Olivucci, M. A. Robb, Chem. Soc. Rev. 25, 321 (1996).
V. Sidis, Adv. Chem. Phys. 82, 73 (1992).
J. J. Kaufman, Adv. Chem. Phys. 28, 113 (1975).
Fitting to analytic functions is reviewed in
A. D. Isaacson, J. Phys. Chem. 96, 531 (1992).
W. C. Ermler, H. C. Hsieh, Advances in Molecular Electronic Structure Theory T. H. Dunning, Jr., Ed., 1, JAI, Greenwich (1990).
G. C. Schatz, Advances in Molecular Electronic Structure Theory T. H. Dunning, Jr., Ed., 85, JAI, Greenwich (1990).
The DIM method is reviewed in
J.C. Tully, Potential Energy Surfaces K. P. Lawly, Ed., 63 John Wiley & Sons, New York (1980).
Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems. David C. Young Copyright ( 2001 John Wiley & Sons, Inc.
ISBNs: 0-471-33368-9 (Hardback); 0-471-22065-5 (Electronic)
21 Conformation Searching
Geometry optimization methods start with an initial geometry and then change that geometry to ®nd a lower-energy shape. This usually results in ®nding a local minimum of the energy as depicted in Figure 21.1. This local minimum corresponds to the conformer that is closest to the starting geometry. In order to ®nd the most stable conformer (a global minimum of the energy), some type of algorithm must be used and then many di¨erent geometries tried to ®nd the lowest-energy one.
For any given molecule, this is important because the lowest-energy conformers will have the largest weight in the ensemble of energetically accessible conformers. This is particularly important in biochemical research in order to determine a protein structure from its sequence. A protein sequence is much easier to determine than an experimental protein structure by X-ray di¨raction or neutron di¨raction. Immense amounts of work have gone into attempting to solve this protein-folding problem computationally. This is very di½cult due to the excessively large number of conformers for such a big molecule. This problem is complicated by the fact that even molecular mechanics calculations require a ®nite amount of computer time, which may be too long to spend on each trial conformer of a large protein.
Which conformation is most important falls into one of three categories. First, simply asking what shape a molecule has corresponds to the lowestenergy conformer. Second, examining a reaction corresponds to asking about the conformer that is in the correct shape to undergo the reaction. Since the di¨erence in energy between conformers is often only a few kcal/mole, it is not uncommon to ®nd reactions in which the active conformer is not necessarily the lowest-energy one. Third, predicting an observable property of the system may require using the statistical weights of that property for all the energetically accessible conformers of the system.
Conformation search algorithms are an automated means for generating many di¨erent conformers and then comparing them based on their relative energies. Due to the immensely large number of possible conformers of a large molecule, it is desirable to do this with a minimum amount of CPU time. Quite often, all bond lengths are held ®xed in the course of the search, which is a very reasonable approximation. Frequently, bond angles are held ®xed also, which is a fairly reasonable approximation.
Algorithms that displace the Cartesian coordinates of atoms have also been
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180 21 CONFORMATION SEARCHING
local minimum
global minimum
Energy
optimization starting here leads to the local minimum
Conformation space
FIGURE 21.1 A one-dimensional representation of the energy of all possible conformers of a simple molecule.
devised. This is usually one of the less e½cient methods for exploring conformers of ¯exible molecules. It can also have the unintended side e¨ect of changing the stereochemistry of the compound. This can be an e½cient way to ®nd conformers of nonaromatic ring systems.
From geometry optimization studies, it is known that energy often decreases quite rapidly in the ®rst few steps of the optimization. This is particularly common when using conjugate gradient algorithms, which are often employed with molecular mechanics methods. Because of this, geometry minimization is sometimes incorporated in conformation search algorithms. Some algorithms do one or two minimization steps after generating each trial conformer. Other algorithms will perform a minimization only for the lowest-energy conformers. It is advisable to do an energy minimization for a number of the lowest-energy structures found by a conformation search, not just the single lowest-energy structure.
The sections below describe the most commonly used techniques. There are many variations and permutations for all of these. The reader is referred to the software documentation and original literature for clari®cation of the details.
21.1GRID SEARCHES
The simplest way to search the conformation space is by simply choosing a set of conformers, each of which is di¨erent by a set number of degrees from the previous one. If each bond is divided into M di¨erent angles and N bonds are
21.1 GRID SEARCHES |
181 |
Energy
(a) |
Conformation space |
|
Energy
(b) |
Conformation space |
FIGURE 21.2 Sampling of conformation space using a grid search. (a) Sampling with a ®ne grid. (b) Sampling with a coarse grid.
rotated, this results in M N possible conformers. In order to search the conformation space adequately, these points must be fairly close together as depicted in Figure 21.2. Although very slow, a grid search with a fairly small step size is the only way to be completely sure that the absolute global minimum has been found. The number of steps can be best limited by checking only the known staggered conformations for each bond, but this can still give a large number of conformers.