Derived Molecular Properties

Some simple size indicators for a molecule are the total number of atoms (N/J, the molecular mass (W m) th total number of valence electrons (Zy). The molecular 

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Apart from serving to partition crystallographically measured electron densities, the atoms-in-molecules procedure is also used to derive molecular properties from isolated-molecule densities, calculated by ab initio methods. Discrepancies between sum-over-atoms and whole-molecule calculations,... 

Beck, B., Glen, R.C. and Clark, T. (1996). The Inhibition of Alpha Chymotrypsin Predicted Using Theoretically Derived Molecular Properties. J.MoLGraphics, 14,130. 

The molecular g-values and susceptibility anisotropies are fitted to the observed multiplet splittings by a least squares procedure and the results are used to calculate derived molecular properties such as the molecular electric quadrupole moments. 

Schroeder et al (Refs 4—6 8), as a corollary to the study of Tarver et al, have proceeded with a research program in quantum chemistry in an attempt to relate the ballistic and expl properties of compds such as the tetrazoles with their calcd and exptly derived molecular properties and to understand the chemistry of these compds. They reported (ca 1975) progress in their study in that the apparent greater electron withdrawing ability of the 1- rather than the 2- substituted 5-tetrazolyl groups is ... due to both resonance and electrostatic field effects. .. (Ref 8, p 19)... 

A synopsis of ab initio calculations presenting the methods for computing wave functions and the derived molecular properties is given in Table 5. References for calculations with semiempirical methods are briefly given below. The abbreviations and symbols used are explained on p. 235. 

TTie CNDO/2 method [46] of Pople et al. was chosen for these types of studies because its computational simplicity allows the feasible consideration of large, complex, and conformationally mobile systems such as cyclodextrin host-guest systems. In contrast to ab initio methods, which seek to directly derive molecular properties from the basic principles of quantum mechanics, CNDO/2 is a semiempirical method, which means that not only are the expressions for wavefunctions simplified algebraically, but also that certain coefficients associated with the resulting approximate functions are arbitrarily assigned in order to duplicate experimental data. 

Reading this book you will not only learn how the most important molecular properties are defined but will also learn how to derive molecular properties for new experimental setups. Furthermore, you can understand the relations between various molecular properties and how this can be used to predict the outcome of one experiment based on other measurements. In the third part of the book you acquire a thorough understanding of quantum chemical methods for the calculation of molecular properties. In particular, you find out how the various quantum mechanical methods are related to each other. At the same time you will become acquainted with different techniques for deriving computational methods and will learn how to apply these techniques to different types of wavefunctions. This will allow you to derive new methods on your own. 

The P matrix obtained from fragment calculations may be used to derive molecular properties from the density matrix using the standard rules of quantum mechanics. 

In equilibrium statistical mechanics, one is concerned with the thennodynamic and other macroscopic properties of matter. The aim is to derive these properties from the laws of molecular dynamics and thus create a link between microscopic molecular motion and thennodynamic behaviour. A typical macroscopic system is composed of a large number A of molecules occupying a volume V which is large compared to that occupied by a molecule ... 

J0rgensen P and Simons J (eds) 1986 Geometrical Derivatives of Energy Surfaces and Molecular Properties (Boston, MA Reidel)... 

In fact, there is a hierarchy in calculating molecular properties by additivity of atomic, bond, or group properties, as was pointed out some time ago by Benson [1, 2]. The larger the substructures that have to be considered, the larger the number of inaements that can be derived and the higher the accuracy in the values obtained for a molecular property. 

The next step towards increasing the accuracy in estimating molecular properties is to use different contributions for atoms in different hybridi2ation states. This simple extension is sufficient to reproduce mean molecular polarizabilities to within 1-3 % of the experimental value. The estimation of mean molecular polarizabilities from atomic refractions has a long history, dating back to around 1911 [7], Miller and Sav-chik were the first to propose a method that considered atom hybridization in which each atom is characterized by its state of atomic hybridization [8]. They derived a formula for calculating these contributions on the basis of a theoretical interpretation of variational perturbation results and on the basis of molecular orbital theory. 

The one-center two-electron integrals in the MNDO method are derived from experimental data on isolated atoms. Most were obtained from Oleari s work L. Oleari, L. DiSipio, and G. DeMich-ells. Mol. Phys., 10, 97( 1977)1, but a few were obtained by IDewar using fits to molecular properties. 

The correct treatment of boundaries and boundary effects is crucial to simulation methods because it enables macroscopic properties to be calculated from simulations using relatively small numbers of particles. The importance of boundary effects can be illustrated by considering the following simple example. Suppose we have a cube of volume 1 litre which is filled with water at room temperature. The cube contains approximately 3.3 X 10 molecules. Interactions with the walls can extend up to 10 molecular diameters into the fluid. The diameter of the water molecule is approximately 2.8 A and so the number of water molecules that are interacting with the boundary is about 2 x 10. So only about one in 1.5 million water molecules is influenced by interactions with the walls of the container. The number of particles in a Monte Carlo or molecular dynamics simulation is far fewer than 10 -10 and is frequently less than 1000. In a system of 1000 water molecules most, if not all of them, would be within the influence of the walls of the boundary. Clecirly, a simulation of 1000 water molecules in a vessel would not be an appropriate way to derive bulk properties. The alternative is to dispense with the container altogether. Now, approximately three-quarters of the molecules would be at the surface of the sample rather than being in the bulk. Such a situation would be relevcUit to studies of liquid drops, but not to studies of bulk phenomena. 

Another way of obtaining molecular properties is to use the Hellmann-Feynman theorem. This theorem states that the derivative of energy with respect to some property P is given by... 

When structure-property relationships are mentioned in the current literature, it usually implies a quantitative mathematical relationship. Such relationships are most often derived by using curve-fitting software to find the linear combination of molecular properties that best predicts the property for a set of known compounds. This prediction equation can be used for either the interpolation or extrapolation of test set results. Interpolation is usually more accurate than extrapolation. 

PM3, developed by James J.P. Stewart, is a reparameterization of AMI, which is based on the neglect of diatomic differential overlap (NDDO) approximation. NDDO retains all one-center differential overlap terms when Coulomb and exchange integrals are computed. PM3 differs from AMI only in the values of the parameters. The parameters for PM3 were derived by comparing a much larger number and wider variety of experimental versus computed molecular properties. Typically, non-bonded interactions are less repulsive in PM3 than in AMI. PM3 is primarily used for organic molecules, but is also parameterized for many main group elements. 

When you perform a single point semi-empirical or ab initio calculation, you obtain the energy and the first derivatives of the energy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a number of other molecular properties that could be available to you. Molecular properties are basically an average over the wave function of certain operators describing the property. For example, the electronic dipole operator is basically just the operator for the position of an electron and the electronic contribution to the dipole moment is... 

Molecular Dynamics and Monte Carlo Simulations. At the heart of the method of molecular dynamics is a simulation model consisting of potential energy functions, or force fields. Molecular dynamics calculations represent a deterministic method, ie, one based on the assumption that atoms move according to laws of Newtonian mechanics. Molecular dynamics simulations can be performed for short time-periods, eg, 50—100 picoseconds, to examine localized very high frequency motions, such as bond length distortions, or, over much longer periods of time, eg, 500—2000 ps, in order to derive equiUbrium properties. It is worthwhile to summarize what properties researchers can expect to evaluate by performing molecular simulations ... 

This is not a unique way of classifying molecular properties. For example, Dykstra et al. (1990) concentrate on the response of a system to an apphed external field the electric dipole moment can be defined as the first derivative of the energy with respect to the field, and so on. I will stick with the Boys and Cook nomenclature as a broad basis for discussion. 

Mixed derivatives refer to cross terms if the energy is expanded in more than one perturbation. There are many such mixed derivatives which translate into molecular properties, below are a few examples. 

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