Quantum Hamiltonian, classical

The location of the quantum/classical boundary across a covalent bond also has implications for the energy terms evaluated in the Emm term. Classical energy terms that involve only quantum atoms are not evaluated. These are accounted for by the quantum Hamiltonian. Classical energy terms that include at least one classical atom are evaluated. Referring to Figure 2, the Ca—Cp bond term the N — Ca—Cp, C — Ca—Cp, Ha— Ca—Cp, Ca—Cp — Hpi, and Ca—Cp — Hp2 angle tenns and the proper dihedral terms involving a classical atom are all included. 

To begin with, we compare the stepsizes used in the simulations (Fig. 3). As pointed out before, it seems to be unreasonable to equip the Pickaback scheme with a stepsize control, because, as we indeed observe in Fig. 3, the stepsize never increases above a given level. This level depends solely on the eigenvalues of the quantum Hamiltonian. When analyzing the other integrators, we observe that the stepsize control just adapts to the dynamical behavior of the classical subsystem. The internal (quantal) dynamics of the Hydrogen-Chlorine subsystem does not lead to stepsize reductions. 

For QM-MM methods it is assumed that the effective Hamiltonian can be partitioned into quantum and classical components by writing [9]... 

The critical points of the equivalent classical Hamiltonian occur at stationary state energies of the quantum Hamiltonian H and correspond to stationary states in both the quantum and generalized classical pictures. These points are characterized by the constrained generalized eigenvalue equation obtained by setting the time variation to zero in Eq. (4.17)... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

The inter/intramolecular potentials that have been described may be viewed as classical in nature. An alternative is a hybrid quantum-mechanical/classical approach, in which the solute molecule is treated quantum-mechanically, but interactions involving the solvent are handled classically. Such methods are often labeled QM/MM, the MM reflecting the fact that classical force fields are utilized in molecular mechanics. An effective Hamiltonian Hefl is written for the entire solute/solvent system ... 

Another ambiguity in defining the classical mapping Hamiltonian is related to the fact that different bosonic quantum Hamiltonians may correspond to the same original quantum Hamiltonian H. This problem was already discussed in Section VI.A.2 for A-level systems. In the context of nonadiabatic dynamics, a different version of the mapping Hamiltonian is given by... 

In cases where the classical energy, and hence the quantum Hamiltonian, do not contain terms that are explicitly time dependent (e.g., interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms discussed later in this text), the separations of variables techniques can be used to reduce the Schrodinger equation to a time-independent equation. 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

G. Casati, B. Chirkov, J. Ford, and F.M. Izrailev, in G. Casati and J. Ford, (Eds.), Stochastic Behavior in Classical and Quantum Hamiltonian Systems of Lecture Notes in Physics, Vol. 93, Springer, Berlin, 1979. 

The quantum Hamiltonian of the classical kicked rotor, defined by the classical Hamiltonian function (5.1.2), is easily obtained by canonical quantization. On replacing the classical angular momentum L by the quantum angular momentum operator L according to... 

Casati, G., Chirikov, B.V., Izraelev, F.M. and Ford, J. (1979). Stochastic behavior of a quantum pendulum under a periodic perturbation, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, eds. G. Casati and J. Ford (Springer, New York). 

It should however be noted that the eigenfunctions of the hamiltonian are not eigenfunctions of the linear momentum operator. Accordingly, a measurement of the momentum does not lead to p = 2mE , but to a probability distribution as shown in Fig. 2.3 (refs. 18 and 19). It can be noted that the most probable momentum is not p , when the particle is in a state except when n becomes large. Then the quantum and classical descriptions are similar. 

The third method is a direct finite-size scaling approach to study the critical behavior of the quantum Hamiltonian without the need to make any explicit analogy to classical statistical mechanics [54,88]. The truncated wave function that approximate the eigenfunction Eq. (52) is given by... 

The validity of this mixed quantum-classical scheme is far from obvious, and important questions regarding its applicability may be raised. For example, does this coupled system of quantum and classical degrees of freedom conserve the total energy as it should (the answer is a qualified yes it may be shown that the sum of the energy of the classical system and the instantaneous expectation value of the Hamiltonian of the quantum system is a constant of motion of this dynamics). Experience shows that in many cases this scheme provides a good approximation, at least for the short-time dynamics. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

Computer simulation can employ both quantum and classical mechanical Hamiltonians to study the structure, function, and dynamics at the atomic... 

Many important problems in computational physics and chemistry can be reduced to the computation of dominant eigenvalues of matrices of high or infinite order. We shall focus on just a few of the numerous examples of such matrices, namely, quantum mechanical Hamiltonians, Markov matrices, and transfer matrices. Quantum Hamiltonians, unlike the other two, probably can do without introduction. Markov matrices are used both in equilibrium and nonequilibrium statistical mechanics to describe dynamical phenomena. Transfer matrices were introduced by Kramers and Wannier in 1941 to study the two-dimensional Ising model [1], and ever since, important work on lattice models in classical statistical mechanics has been done with transfer matrices, producing both exact and numerical results [2]. 

The extension of the previous treatment to a system in a medium starts from the L-vN equation for t t) to derive the equation of motion of the RDOp p for the primary (or p-) region, that includes a dissipative term, given by the LiouviUe superoperator (t). This describes the interaction with a secondary (or s-) region. The PWT is applied only to the p-region so that the quantum and classical Hamiltonians of the previous section refer to the p-region the s-region is described in terms of its collective motions and a distribution of initial motion amplitudes [9,14]. 

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