The Modeling of Molecules Through Computational Methods
The computations of molecular properties using the common computational techniques are explained in this chapter. The chapter includes the principle of optimization, specifically mentioning gradient-based methods. Various computational requirements such a
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The Modeling of Molecules Through Computational Methods
12.1 Introduction Performing a geometry optimization is the primary step in studying a molecule using computational techniques. Geometry optimizations classically attempt to locate a minimum on the potential energy surface in order to foretell the equilibrium structures of molecular systems. They may also be used to locate transition structures or intermediate structures. Moreover, the geometry of a molecule determines many of its physical and chemical properties. We know that the energy of a molecule changes with its structure. Hence, understanding the methods of geometry optimization is the major requirement for energy minimization. It is essential to understand the geometry of a molecule before running computations.
12.2 Optimization Optimization modeling can be carried out by identifying the objectives, the design variables, and the constraints, and by using an algorithm to find the solution to the problem. Optimality conditions will help us to determine whether we have indeed reached our goal of an optimum solution.
12.2.1 Multivariable Optimization Algorithms Optimization problems, which we come across in molecular modeling, are multivariable problems, where the objective functions have more than a single variable on which the given function depends on. If we consider a “two variable problem”, say, f (x) = x21 + x22 , and say x1 = 3 and x2 = 4, then every x1 and x2 has a function value (i.e., height). This function can be represented by a surface (Fig. 12.1). We
K. I. Ramachandran et al., Computational Chemistry and Molecular Modeling DOI: 10.1007/978-3-540-77304-7, ©Springer 2008
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12 The Modeling of Molecules Through Computational Methods
Fig. 12.1 Quadratic form of a function
have to find the minimum value of the function and at what values of x1 and x2 is the minimum value attained. For example, the minimum occurs at f (x) = 0 and occurs when x1 = x2 = 0. We can also put constraints such as x1 + x2 = 5, in which case the solution must lie on the line of constraint. However, we will be discussing only unconstrained problems now.
12.2.2 Level Sets, Level Curves, and Gradients The function values under study are represented as contour maps with circles representing each function value (Fig. 12.2). Any function f (x) = C is a level set, which is a set of points having the same height. These contours are called level sets or level curves. At any point on the circle or curve, the function value will be the same (Fig. 12.3). The outermost contour will have the highest function value and the inner circles will progressively have smaller and smaller values. At the bottommost point, the function will have zero value and is said to be the minimum at that point. At each point on the curve, there are gradients, given by ∇ f (x), pointing to the steepest direction. The direction of steepest descent is given by −∇ f (x), which we get by searching in the opposite direction. The contour map is a vector field, with gradients at every point. The gradie
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