The Modified Hamiltonian

We begin by asking the following question which takes us right to the heart of what is needed in molecular dynamics how is the energy affected by numerical discretization  

Let us begin with an illustrative example. Consider the harmonic oscillator with frequency which has Hamiltonian H q,p) = and consider the 

If H(q,p) = E then, for typical steps, we cannot expect H(Q, P) = E. (Just insert the formulas for Q and P into the Hamiltonian and check that the value is not the same as H q,p).) 

This means that W is a conserved quantity of the numerical method. 

For some values of the stepsize, it is possible that the points generated by the numerical method will constitute a finite periodic sequence. For example let = 1 and use the large stepsize of h = 1. Then from the initial point (qo,po) = (1,0) we have qi = qo + po = 1, and pi = po — qi = —1, so we visit (1,-1), then (0, —1), then (—1,0), (-1,1), (0,1) and finally return to (1,0) (diagrammed in the left panel of Fig. 3.1). For a different choice of the stepsize, the point sequence may 

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Ehrenfest trajectory for three-dimensional D + H2 generated by the RWP method, that is, the modified Hamiltonian operator f H). Dotted curves in (b) correspond to the Ehrenfest trajectory determined by the usual Schrddinger equation. See text for further details. 

In order to be able to determine re we must consider rotations. A straightforward procedure is a generalization of that discussed in the previous section, and it is called the cranking method. The method has been used extensively in nuclear physics (Schaaser and Brink, 1984). It consists in evaluating the expectation value of the modified Hamiltonian... 

Let us treat the proton dynamics starting from the modified Hamiltonian (153), that is [149,150],... 

If the species is charged then an appropriate Bom term must also be added. The reaction field model can be incorporated into quantum mechanics, where it is commonly referred to as the self-consistent reaction field (SCRF) method, by considering the reaction field to be a perturbation of the Hamiltonian for an isolated molecule. The modified Hamiltonian of the system is then given by ... 

For high precision calculations, and especially for the isotope shift, it is necessary to include also the motion of the nucleus in the center-of-mass (CM) frame. A transformation to CM plus relative coordinates yields the additional — fx./M)V V2 mass polarization term in the modified Hamiltonian... 

To cope with this inconsistency, we introduce the modified Hamiltonian (operator) matrix for the case wq 0, cf. Eq. (1.14),... 

As we have seen in the Chap. 3, when a symplectic method is applied to a molecular dynamics problem it induces a perturbed Hamiltonian (energy) function. For the Verlet scheme the modified Hamiltonian is... 

Note that when a symplectic integration method is used, we have, from the discussion in Chap. 3, a perturbed energy function and, moreover, from Theorem 3.1, the error in energy H is OQf), nonetheless the perturbations are large for a large stepsize h. In the Shadow Hybrid Monte-Carlo (SHMC) method [2, 3, 188], the accept-reject test is based on the modified Hamiltonian Hh (see Chap. 3), derived from the Baker-Campbell-Hausdorff expansion. SHMC can improve efficiency by decreasing the rejection rate. 

Bloch. Numerical details concerning the modified 3D g-HO Hamiltonian are given in Section 6, together with a study of supershells in the firamework of this Hamiltonian. In Section 7 a variational method leading from the usual harmonic oscillator to the Morse oscillator is introduced, while in Section 8 this method is applied for deriving the modified Hamiltonian introduced in Section 5 firom the original 3D g-HO Hamiltonian. Finally in Section 9 a discussion of the present results and plans for future work are given. 

Derivation of the modified Hamiltonian for the 3D -HO through the variational method... 

This transformation has to be applied to the left and the right of the Dirac Hamiltonian to obtain the modified Hamiltonian. The same applies to operators for various molecular properties, which must also be modified the unmodified form is simply multiplied on the left and the right by the transformation operator T to obtain the modified form. We consider here both an operator defined by a scalar potential and an operator defined by a vector potential. 

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