Hamiltonian simulation

Answer The authors did indeed evaluate the Mossbauer data by spin Hamiltonian simulations assnming an 5 = 2 as well as an 5 = 1 gronnd state. In this way they conld dednce two sets of hyperfine and fine stmctme parameters simnlations assnming an A = 1 gronnd state yield an axial hyperfine tensor A = -33.3 T. 

Figure 15 (a) The 4.2 K Mossbauer spectra of [(Fe(IV)=0)(TMC)(NCCH3)](0Tf)2 in acetonitrile recorded in (A) zero field and (B)a parallel field of 6.5 T. The solid line represents a spin Hamilton simulation with the parameters described in the text, (b) Mossbauer spectra of [Fe(lV)=(0)(TMCS)] recorded at temperatures and applied fields that are indicated. The solid lines represent spin Hamiltonian simulations with parameters described in the text. The spectra were simulated in the slow (at 4.2 K) and fast (at 30 K) spin fluctuation limit. The applied field was directed parallel to the observed y radiation. The doublet drawn above the topmost experimental spectrum (0 T, 4 K) represents a 7% Fe(ll) contribution from the starting complex. (From J. U. Rohde et al. (2003) Science 299 1037-1039. Reprinted with permission from AAAS)... 

Figure 8 Mossbauer spectrum of a frozen aqueous solution of [ Fe +]-ferrioxamine B (12mM) employing BSA (lOOmM) as a dilutant to minimize spin-spin relaxation. The solid line represents a simulation based on a spin Hamiltonian line width = 0.35 mm s zero-field splitting, D = 1.2 cm E rhombicity parameter, E/D = 0.33 8 = 0.52mms A q = —0.84mms asymmetry parameter, rj = and isotropic hyperfine coupling tensor Axx/gN/XN = Ayy/gNMN = Azz/gx/XN = —22.1 T. The simulation does not completely fit the experimental data. This discrepancy is caused by relaxation effects that are not dealt with in the spin Hamiltonian simulation...

Figure 12 Field-dependent Mossbauer spectra of the ferric low-spin heme complex [TPPFe(NH2PzH)2]Cl. The solid lines are spin Hamiltonian simulations for 5=1 /2 with parameters discussed in the text and given in Ref. 42 (From Schiinemann et With kind permission from Springer Science Business Media)...

Perturbation theory is also used to calculate free energy differences between distinct systems by computer simulation. This computational alchemy is accomplished by the use of a switching parameter X, ranging from zero to one, that transfonns tire Hamiltonian of one system to the other. The linear relation... 

For some systems qiiasiperiodic (or nearly qiiasiperiodic) motion exists above the unimoleciilar tlireshold, and intrinsic non-RRKM lifetime distributions result. This type of behaviour has been found for Hamiltonians with low uninioleciilar tliresholds, widely separated frequencies and/or disparate masses [12,, ]. Thus, classical trajectory simulations perfomied for realistic Hamiltonians predict that, for some molecules, the uninioleciilar rate constant may be strongly sensitive to the modes excited in the molecule, in agreement with the Slater theory. This property is called mode specificity and is discussed in the next section. 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

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. 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

If X and Y do not overlap in phase space then the value of the free energy differei calculated using Equation (11.6) will not be very accurate, because we will not adequat sample the phase space of Y when simulating X. This problem arises when the enei difference between the two states is much larger than k T . y - x 3> kgT. How tl can we obtain accurate estimates of the free energy difference under such circumstanc Consider what happens if we introduce a state that is intermediate between X and with a Hamiltonian and a free energy A(l) ... 

Tlierc are two major sources of error associated with the calculation of free energies fi computer simulations. Errors may arise from inaccuracies in the Hamiltonian, be it potential model chosen or its implementation (the treatment of long-range forces, e j lie second source of error arises from an insufficient sampling of phase space. 

Note that Eq. (40d), in fact, is redundant, because the other three equations form a closed set. Nonetheless, if we solve the equations of motion for s as well, we can use the following, as a diagnostic tool, because this quantity has to be conserved during the simulation even though //nos6 is no longer a Hamiltonian ... 

Other quantum simulations involve simulations with effective Hamiltonians [261-263] or the simulation of ground state wave properties by Green s function Monte Carlo or diffusion Monte Carlo for reviews and further references on these methods see Refs. 162, 264-268. 

Path integral Monte Carlo simulations were performed [175] for the system with Hamiltonian (Eq. (25)) for uj = ujq/J = A (where / = 1) with N = 256 particles and a Trotter dimension P = 64 chosen to achieve good computer performance. It turned out that only data with noise of less than 0.1% led to statistically reliable results, which were only possible to obtain with about 10 MC steps. The whole study took approximately 5000 CPU hours on a CRAY YMP. 

The properties of the Hamiltonian (41) have been studied by means of PIMC simulations [327,328]. We stress here that for rotational motion in... 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

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