Hamiltonian energy function

Thus, the solution to the classical problem with the Hamiltonian (energy) function given by Equation 3.19 is a set of 3N harmonic oscillators with 3N frequencies Aj = 4tt2v2. The A s result from the diagonalization of the F matrix of Equation 3.15. 

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... 

Let a system be provided with Hamiltonian (energy) function H = H z), z e M", and microcanonical dynamics... 

Hamiltonian Energy function of a system expressing the total energy of the system as a function of the coordinates and momenta of its particles, molecules, and/or ions. 

The Hamiltonian energy function of the protein is Ho r, p). The variable w can be added with an associated fictitious mass m so that the transformed Hamiltonian is... 

A classical system is described by a classical Hamiltonian, H, which is a function of both coordinates r and momenta p. For regular molecular systems, where the potential energy function is independent of time and velocity, the Hamiltonian is equal to the total energy. 

The first Hamiltonian was used in the early simulations on two-dimensional glass-forming lattice polymers [42] the second one is now most frequently used in two and three dimensions [4]. Just to illustrate the effect of such an energy function, which is given by the bond length, Fig. 10 shows two different states of a two-dimensional polymer melt and, in part. 

Many second-order reversible rules of the above form allow a pseudo-Hamiltonian prescription. The evolution of such systems may then be defined as any configurational change that conserves an energy function . We discuss this Hamiltonian formulation a bit later in this section. 

As mentioned above, Hopfield s original approach to this problem was to introduce an energy function reminiscent of a spin-glass Hamiltonian ... 

Eh Linear energy functional based on the Hamiltonian H it acts on the space of A-particle Trace Class operators. 

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... 

Since the Coulomb, exchange, and correlation energies are all consequences of the interelectronic 1 /r12 operator in the Hamiltonian, one can define the exchange energy functional Ex [p] in the same manner as... 

The potential energy functions are expressed in terms of q, 0 = 1,2,. 6), which explicitly exhibits its independence of the coordinates of the center of mass. Again, since the momenta conjugate to coordinates 0/7. q, qg), i.e. Pi,p andp9 remains constants of motion during the entire collision, the term containing them in the Hamiltonian has been subtracted. 

So Equation 3.19 shows the Hamiltonian, the total energy function of classical mechanics, is the sum of 3N terms each of the form... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

In this paper we will not pursue such formal developments any further, and instead use mean field ideas and heuristic arguments to motivate the choice of the appropriate free energy functional. We represent the intrinsic free energy functional in the form of an effective 2D step Hamiltonian H and imagine on physical grounds that it has the... 

The first-order correction can be thought of as arising from the response of the wavefunction (as contained in its LCAO-MO and Cl amplitudes and basis functions %v) plus the response of the Hamiltonian to the external field. Because the MCSCF energy functional has been made stationary with respect to variations in the Cj and Cj a amplitudes, the second and third terms above vanish ... 

Thus it is necessary to study the stability of the state in the G well. To obtain the energy functional it is convenient to start from the Hamiltonian given by Eqs. 1-3. Assuming a stationary solution ijf t)=il/ exp -iEt/1t), and substituting Eq. 5 into the Hamiltonian, we obtain the energy functional... 

---