Quantum Hamiltonian computing

Keywords Intramolecular kirchhoff laws Molecular electronics Molecular logic gates Single molecule electronic circuits Quantum hamiltonian computing... 

A more elegant model of quantum mechanical computers was introduced by Feynman ([feyii85], [feynSb]). Although the formalism is similar to Benioff s - both approaches seek to define an appropriate quantum mechanical Hamiltonian H whose time evolution effectively represents the execution of a desired computation... 

The two avenues above recalled, namely ab-initio computations on clusters and Molecular Dynamics on one hand and continuum model on the other, are somewhat bridged by those techniques where the solvent is included in the hamiltonian at the electrostatic level with a discrete representation [13,17], It is important to stress that quantum-mechanical computations imply a temperature of zero K, whereas Molecular Dynamics computations do include temperature. As it is well known, this inclusion is of paramount importance and allows also the consideration of entropic effects and thus free-energy, essential parameters in any reaction. 

Three known facts are essentially important in the development of a divide-and-conquer strategy. First, the KS Hamiltonian is a single particle operator that depends only on the total density, not on individual orbitals. This enables one to project the energy density in real space in the same manner in which one projects the density (see below). Second, any complete basis set can solve the KS equation exactly no matter where the centers of the basis functions are. Thus, one has the freedom to select the centers. It is well known that for a finite basis set the basis functions can be tailored to better represent wavefunctions, and thus the density, of a particular region. The inclusion of basis functions at the midpoint of a chemical bond is the best known example. Finally, the atomic centered basis functions used in almost all quantum chemistry computations decay exponentially. Hence both the density and the energy density contributed by atomic centered basis functions also decrease rapidly. All these... 

DK approximation and it will be shown that the result is independent of the chosen parametrisation. This approach has not been investigated in the literature so far. We will denote the resulting operator equations as the generalised Douglas-Kroll transformation. We conclude this section by a presentation of some technical aspects of the implementation of the DK Hamiltonian into existing quantum chemical computer codes. 

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 Hartree-Fock theory is the starting point for many high level quantum chemical computations of molecules. Hence it is rather well presented in standard sources for quantum chemistry.We shall therefore discuss only the bare essence of the theory but we shall do so in such a manner that will allow us to rapidly arrive at a Hamiltonian that contains the essence of the physics we want to describe. 

Consider a quantum mechanical description of the hydrogen molecular ion in its simplest version. Let us use the molecular orbital theory with the atomic basis set composed of only two Slater Type Orbitals (STOs) Isa and Is , centered on the nuclei a and b. The mean value of the Hamiltonian computed with the bonding (+) and antibonding (—) orbital (see p. 439 and Appendix D available at booksite.elsevier.com/978-0-444-59436-5) reads as... 

As seen in second part of these thesis, the development of efficient quantum dynamical computational methods also opened the door to the study of the laser control of polyatomic molecular systems. In a first application, we used a simplified model Hamiltonian describing the excited state dynamics of pyrazine to investigate the laser control of its ultrafast radiationless decay by a strong non-resonant laser... 

As implied by Equation (4.11) and Equation (4.12), the computational cost of LR-TDDFT grows rapidly with increasing system size, because the size of the response matrices is proportional to the product of the numbers of occupied and unoccupied molecular orbitals, that is, (Voce x Aunocc- Moreover, the diffuse unoccupied orbitals usually require very large basis sets to satisfactorily converge the results. To circumvent these numerical challenges, a RT-TDDFT method was derived to track the response of an optical system upon external perturbation. In RT-TDDFT, an additional term, E r) x ]i r), is added to the system s quantum Hamiltonian, Hq, to reflect Kght-matter interactions ... 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

The use of QM-MD as opposed to QM-MM minimization techniques is computationally intensive and thus precluded the use of an ab initio or density functional method for the quantum region. This study was performed with an AMi Hamiltonian, and the first step of the dephosphorylation reaction was studied (see Fig. 4). Because of the important role that phosphorus has in biological systems [62], phosphatase reactions have been studied extensively [63]. From experimental data it is believed that Cys-i2 and Asp-i29 residues are involved in the first step of the dephosphorylation reaction of BPTP [64,65]. Alaliambra et al. [30] included the side chains of the phosphorylated tyrosine, Cys-i2, and Asp-i 29 in the quantum region, with link atoms used at the quantum/classical boundaries. In this study the protein was not truncated and was surrounded with a 24 A radius sphere of water molecules. Stochastic boundary methods were applied [66]. 

The final application considered in this chapter is chosen to illustrate the application of a QM-MM study of an enzyme reaction that employs an ab initio Hamiltonian in the quantum region [67]. Because of the computational intensity of such calculations there are currently very few examples in the literahire of QM-MM shidies that use a quanhim mechanical technique that is more sopliisticated than a semiempirical method. MuUiolland et al. [67] recently reported a study of part of the reaction catalyzed by citrate synthase (CS) in wliich the quanhim region is treated by Hartree-Fock and MP2 methods [10,51],... 

Benioff introduced a series of Hamiltonians describing the evolution of a system consisting of spin-1/2 particles (spins up/down corresponding to binary logical states 0/1) occupying the sites of a lattice. The initial. state of the system I (t = 0) corresponds to the input state of a computation. Benioff s systems evolve, under the action of a Hamiltonian, in such a way that the quantum states (0),... 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

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