Optimised potential energy functions

The parameter sets PEP301, PEP302, PEP303, PEP304 and PEP401 are documented in chapter 9 They contain parameters for only C and H, and only in saturated compounds. 

In addition to the data in Tables 9-2, 9-3, 9-4, 9-6, 9-7 and 9-9, some supplementary data should be mentioned. 

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This word shall be synonymous with "potential energy function parameters , and its meaning shall be those quantities of a potential energy function which are unchanged from one molecule to another within the same series of minimisation or within one iteration in optimisation. 

A consistent force field is conceptually rather different from a force field. It is a potential energy function whose parameters have been fitted, preferably optimised in some objective way, to reproduce structures of a set of molecules and, through the force field derived for each molecule of the set, molecular spectra. Other pro-... 

The next big step came in 1970 with optimisation on crystal pro-parties. The new potential energy function had the same form as the previous, and many parameters for intra-molecular interactions had changed very little- Angle-torsion cross terms were added, and a Lennard-Jones 9-6 form was used. For some interactions, different parameters were used for methyl and methylene groups. Altogether 29 parameters were employed. 

In the other paper, a completely renewed potential energy function for alkanes and cycloalkanes was optimised. In form it is a GVPP with Morse functions for C-C and C-H bond deformations, and with numerous cross terms. Por non-bonded interactions, Lennard-Jones 9-6 plus Coulomb terms were transferred from previous work d... 

Por some reason, the Lifson group, as many others, prefer the r to other structures as basis for their optimisations rather than a structure type which is independent of temperature as the potential energy function is see also section 2.2 and chapter 8. 

CHARMM is the first program system published which deserves the designation "molecular mechanics , in the sense that it treats static, kinematic and dynamic properties. All other programs available are much more limited in scope. A few warnings for the uninitiated are in place The more complicated a system is, and here I mean only modern well-structured and well-documented systems, the more attention is required to maintain and operate it, and that cannot be left solely to the computer people. Also, know-how does not come by itself or on a tape. For many prospective applications, potential energy function parameters, or even the functions themselves, are lacking, just as for the simpler systems. Parameters must be found by trial and error, as is usual, or, preferably, by optimisation, which cannot be done in CHARMM a program of the consistent force field family is necessary. 

The work shows that the parameter set might be quite a good candidate for optimisation. Another approach might be to use a potential energy function of the form of PEP401 or PEP304, see section 9 2, and optimise simultaneously on gas-phase properties of amines and crystal data of their ammonium salts with simple anions. Some exploratory work already done has been encouraging. 

Let p be a vector whose components are the current values of those potential energy function parameters we want to optimise ... 

The elements of the Jacobian or Z matrix are the partial derivatives of observables with respect to potential energy function parameters. Some classes of them are rather laborious to obtain this Is the price we have to pay for being able to optimise simultaneously on many classes of observables. 

The choice of Internal coordinates as an object for optimisation Is obvious use of rotational constants maybe less so. They certainly do not give very detailed Information about the conformation of a molecule, but they are the primary structural Information derived from rotational and ro-vlb spectroscopy on small molecules. The Inclusion of dipole moments Is a must when Coulomb terms are present In the potential energy function. Charges are Included, although they are not experimentally observable quantities, because It may be desirable to lock a parameter set to data derived from photoelectron spectroscopy or from ab Initio calculations with a large basis set. Quite naturally we want to optimise on vibrational spectra, and we shall see below that It Is a bit more cumbersome In the consistent force field context than In traditional normal coordinate analysis. 

Warshel restrict, if my interpretation is correct, the deviation in a difference between two molecules to a deviation in one of them. Finally, the following point of view should be considered There should be something left out from optimisation, to see whether a final potential energy function is not only internally" consistent, but also consistent with external data. The argument came from Martin Karplus during our many discussions in Rehovot in the last months of 1969. I should add that it did not then coincide with the views of Shneior Lifson, but has been used in recent work. ... 

The interdependence of the potential energy function parameters cannot be judged by inspection of the correlation coefficients as is usual in optimisation because of the methods employed here the modified Z matrix is not diagonalised. For the same reason, an estimate of the standard deviation is not possible either ... 

For the purpose of optimising a potential energy function for transition metal complexes with coordinated diamines, another main activity of our group, we have selected a few diamines and diammonium ions for the data bank. 

The two C-C-C-C torsions, in n-butane g and cyclohexane, cannot be compared in this way. They are very important in optimisation especially for the determination of non-bonded potential energy function parameters. 

A few notes on how the selected data are used in optimisation are relevant. As an example, let us look at the C-C bond in alkanes and cycloalkanes. The constancy of the C-C bond length does not mean that we have here a good value for the potential energy function parameter b. The C-C bond length in an actual molecule is in our model determined by other parameters as well, most clearly illustrated in ethane where H-----H non-bonded interactions are equally... 

If we then turn to ethers, we see that the C-C bond is shorter in the normal straight-chain and cyclic ethers than in alkanes. This must be taken care of in the optimisation of potential energy function parameters different parameter values cannot be accepted, if the overall goal of a simple potential energy function for many classes of substances is to be attained. 

The potential energy function with associated parameters called PEP300 was developed by trial and error. It gave quite nice results for many properties of saccharides, see chapter 5, and the alkane part of it was chosen first as a promising initial set of parameters to be optimised. 

This Just shows that the actual analytical form of the non-bonded functions is of no consequence, and that the inclusion of the unit cell of ethane into optimisation has only marginal effect. When the functional form has little or no importance, the simplest can be used, which is the (A,B) form with one-atom parameters. This conclusion is encouraging for later optimisation of potential energy functions which include heteroatoms. 

This chapter will review some of our results obtained on the simpler compounds. I shall describe potential energy functions with parameter sets fitted both by trial and error and by optimisation, and I shall observe chronological order. The older parts include also work on ethers and alcohols. 

A prospective user of a potential energy function Is first of all Interested In Its reliability. I hope to have shown by my examples In sections 9 1 and 9 2 and In this chapter that a parameter set may be good or just good enough for some purposes but Insufficient for others. This applies also to optimised sets and does not mean that a set Is not consistent. Rather, the model behind the potential energy function Is too primitive, or the parametrlsatlon behind the parameter set Is not well chosen for the purpose. 

It is seen from Table 11-2 that the ethane crystal structure is reproduced to within 60 per mille in unit cell dimensions. Note that the two potential energy functions were developed for free molecules. Both will be acceptable candidates for optimisation, but should not be used in general in their present form. 

For optimisation on lattice observables we need partial derivatives of each type of observable with respect to potential energy function parameters. So far, we have two types of observables conformation. Including lattice constants, and lattice energy. 

All members of the consistent force field family of programs are available, from QCPE or from the authors they are described under section 3.4 They are all singular in a way they allow for optimisation of potential energy function parameters on many types of observable, and are therefore splendid tools in the hands of a skilled worker. 

We have optimised the structure of the lowest doublets and quartets of all the possible coordination modes (a-h). PWP and B3LYP functionals give only two minima, corresponding to the q20,0 and p2C,0 coordination modes, in both the doublet and quartet potential energy surfaces (PES). In addition a transition state (TS), corresponding to the g structure, results from the B3LYP computation on doublet states. 

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