Hamiltonian function

III classical MD (cf. textbook [1]) a molecule is modelled by a Hamiltonian function... 

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

Variational calculus, Dreyfus (1962), may be employed to obtain a set of differential equations with certain boundary condition properties, known as the Euler-Lagrange equations. The maximum principle of Pontryagin (1962) can also be applied to provide the same boundary conditions by using a Hamiltonian function. 

The Hamiltonian function for this dynamical problem, using polar coordinates with the polar axis in the direction of the lines of force, is... 

The Hamiltonian function for an electron in a constant magnetic field of strength H parallel to the s axis is1... 

The classical function A is an observable, meaning that it is a physically measurable property of the system. For example, for a one-particle system the Hamiltonian operator H corresponding to the classical Hamiltonian function... 

We next apply these classical relationships to the rigid diatomic molecule. Since the molecule is rotating freely about its center of mass, the potential energy is zero and the classical-mechanical Hamiltonian function H is just the kinetic energy of the two particles,... 

If we restrict our interest to systems for which the potential energy 7 is a function only of the relative position vector r, then the classical Hamiltonian function H is given by... 

In this chapter, we focus on the method of constraints and on ABF. Generalized coordinates are first described and some background material is provided to introduce the different free energy techniques properly. The central formula for practical calculations of the derivative of the free energy is given. Then the method of constraints and ABF are presented. A newly derived formula, which is simpler to implement in a molecular dynamics code, is given. A discussion of some alternative approaches (steered force molecular dynamics [35-37] and metadynamics [30-34]) is provided. Numerical examples illustrate some of the applications of these techniques. We finish with a discussion of parameterized Hamiltonian functions in the context of alchemical transformations. 

For those applications, each system is modeled using different Hamiltonian functions -A and The free energy difference is defined as... 

Several techniques exist to compute AA. Following our earlier discussion for the PMF we will discuss TI. In this approach a parameterized Hamiltonian J a(x, Pa,) iS defined such that when A = 0, Mx = M and when A = 1, Mx = M. MA(x, p ) interpolates smoothly between the two Hamiltonian functions. The free energy A becomes itself a function of A and we have... 

In the previous examples, we considered a parameterized Hamiltonian function and derived equations to compute. 4(A). Let us now consider the dependence of A with temperature. Based on the definition of. 4, we have... 

It is possible to treat the parameter A in the alchemical transformation as a dynamic variable using an extended ensemble [56]. For simplicity of implementation, it has been proposed to use two variables A0 and A i such that A +A = 1. The Hamiltonian function is then defined as [57, 58]... 

The last step consists of proving that ( (a5 F/d ))F = 0 For Hamiltonian systems, it is generally true that the phase-space average of the time derivative of any function of q and p is equal to zero. For an arbitrary function /(q, p) and a Hamiltonian function 34 ... 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

By substituting the equality (A1.7) into the relation (A1.6) and adding the result to Eq. (A1.5), we deduce the Hamiltonian function expressed in terms of normal coordinates ... 

Then in the initial Hamiltonian function (Al.l), all the indices, a and P, of the projections onto the Cartesian axes are the same and thus can be omitted, while the... 

Assume the impurity particle C to be harmonically bound to a main system of oscillators numbered by i = 0, 1, 2,. .. through a single particle with the number i = 0. Fig. A1.5 shows particles labeled by i at cubic lattice sites. The complete Hamiltonian function of the system under discussion is represented as follows ... 

The Lagrangian equations can be turned into another useful form involving generalized coordinates and momenta and by defining the Hamiltonian function... 

In an equivalent classical equation, the variable Ik cancels to give the Hamiltonian function, which for a single particle of mass m,... 

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials... 

Suppose that the system is described by n normal coordinates g. The Hamiltonian function, or energy, is given by... 

The internal energy is, as indicated above, connected to the number of degrees of freedom of the molecule that is the number of squared terms in the Hamiltonian function or the number of independent coordinates needed to describe the motion of the system. Each degree of freedom contributes jRT to the molar internal energy in the classical limit, e.g. at sufficiently high temperatures. A monoatomic gas has three translational degrees of freedom and hence, as shown above, Um =3/2RT andCy m =3/2R. 

In classical mechanics, the Hamiltonian function is the expression of the energy of a molecular system in terms of the momenta of the particles in the system and... 

The interest here is in the energy levels of molecular systems. It is well known that an understanding of these energy levels requires quantum mechanics. The use of quantum mechanics requires knowledge of the Hamiltonian operator Hop which, in Cartesian coordinates, is easily derived from the classical Hamiltonian. Throughout this chapter quantum mechanical operators will be denoted by subscript op . If the classical Hamiltonian function H is written in terms of Cartesian momenta and of interparticle distances appropriate for the system, then the rule for transforming H to Hop is quite straightforward. Just replace each Cartesian momentum component... 

Here H is the Hamiltonian function for one N atomic molecule (i) and s is its symmetry number. One might have expected this result immediately from the Kirkwood formulation for the classical canonical partition function. H is a function of the 3N Cartesian momenta and the 3N Cartesian coordinates of molecule i. 

Let us first consider the relation to the mean-field trajectory method discussed in Section III. To make contact to the classical limit of the mapping formalism, we express the complex electronic variables imaginary parts, that is, mean-field Hamiltonian function which may be defined as... 

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