Molecular system quantum mechanical Hamiltonian

The procedure begins with writing down the quantum mechanical Hamiltonian for a molecular system (electrons + nuclei) in the coordinate space ... 

For both methods, we describe the interactions between the quantum subsystem and the classical subsystem as interactions between charges and/or induced charges/dipoles and a van der Waals term [2-18]. The coupling between the quantum subsystem and the classical subsystem is introduced into the quantum mechanical Hamiltonian by finding effective interaction operators for the interactions between the two subsystems. This provides an effective Schrodinger equation for determining the MCSCF electronic wave function of the molecular system exposed to a classical environment, a structured environment, such as an aerosol particle. 

The stationary points obtained by computational procedures on the PES are for vibrationless molecular systems. The electronic Hamiltonian used in ab initio calculations gives the total electronic energy, Eeiec. A real molecule, however, has vibrational energy even at 0 K, which is the quantum mechanical (QM) zero-point energy (ZPE), l/ihv. At absolute zero, the internal energy, Eo, is defined as the computed electronic energy plus the zero-point energy. 

Traditionally, for molecular systems, one proceeds by considering the electronic Hamiltonian which consists of the quantum mechanical operators for the kinetic energy of the electrons, their mutual Coulombic repulsions, and... 

A Hamiltonian is the quantum mechanical description of an energy contribution. The exact Hamiltonian for a molecular system is ... 

The system studied by use of structurally coupled QM/MM is described with a Hamiltonian comprising of both the quantum mechanically and molecular mechanically treated part. The effective Hamiltonian, Heff, representing the interactions of the whole QM/MM system is ... 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

The system is prepared at t=0 in the quantum state Pik> and the question is how to calculate the probability that at a later time t the system is in the state Fjn>. By construction, these quantum states are solutions of molecular Hamiltonian in absence of the radiation field, Hc->Ho Ho ik> = e k Fik> and H0 Pjn> = Sjn xPJn>. The states are orthogonal. The perturbation driving the jumps between these two states is taken to be H2(p,A)= D exp(icot), where co is the frequency of the incoherent radiation field and D will be a time independent operator. From standard quantum mechanics, the time dependent quantum state is given by ... 

Secondly, information is obtained on the nature of the nuclei in the molecule from the cusp condition [11]. Thirdly, the Hohenberg-Kohn theorem points out that, besides determining the number of electrons, the density also determines the external potential that is present in the molecular Hamiltonian [15]. Once the number of electrons is known from Equation 16.1 and the external potential is determined by the electron density, the Hamiltonian is completely determined. Once the electronic Hamiltonian is determined, one can solve Schrodinger s equation for the wave function, subsequently determining all observable properties of the system. In fact, one can replace the whole set of molecular descriptors by the electron density, because, according to quantum mechanics, all information offered by these descriptors is also available from the electron density. 

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 ... 

The interest here is in the energy levels of molecular systems. It is well known that an understanding of these energy levels requires quantum mechanics. The use of quantum mechanics requires knowledge of the Hamiltonian operator Hop which, in Cartesian coordinates, is easily derived from the classical Hamiltonian. Throughout this chapter quantum mechanical operators will be denoted by subscript op . If the classical Hamiltonian function H is written in terms of Cartesian momenta and of interparticle distances appropriate for the system, then the rule for transforming H to Hop is quite straightforward. Just replace each Cartesian momentum component... 

Fig. 2. The quantum mechanics of the two-state prpblem provide a paradigm for the much more extensive electronic state space of a real molecular or macromolecular system. The eigenvectors c, of the Hamiltonian are symmetric and antisymmetric linear combinations of the localized basis vectors with an eigenvalue splitting of 2A, where s is the overlap integral and A is the direct coupling (the only kind possible in this case)...

Here, for notational convenience, we have assumed that Vnm = We would like to emphasize that the mapping to the continuous Hamiltonian (88) does not involve any approximation, but merely represents the discrete Hamiltonian (1) in an extended Hilbert space. The quantum dynamics generated by both Hamilton operators is thus equivalent. The Hamiltonian (88) describes a general vibronically coupled molecular system, whereby both electronic and nuclear DoF are represented by continuous variables. Contrary to Eq. (1), the quantum-mechanical system described by Eq. (88) therefore has a well-defined classical analog. 

All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist Second, which features of the overall control process are most efficiently subject to feedback control ... 

The application of quantum-mechanical methods to the prediction of electronic structure has yielded much detailed information about atomic and molecular properties.13 Particularly in the past few years, the availability of high-speed computers with large storage capacities has made it possible to examine both atomic and molecular systems using an ab initio variational approach wherein no empirical parameters are employed.14 Variational calculations for molecules employ a Hamiltonian based on the nonrelativistic electrostatic nuclei-electron interaction and a wave function formed by antisymmetrizing a suitable many-electron function of spatial and spin coordinates. For most applications it is also necessary that the wave function represent a particular spin eigenstate and that it have appropriate geometric symmetry. 

It is clear that the various density functional schemes for molecular applications rely on physical aiguments pertaining to specific systems, such as an electron gas, and fitting of parameters to produce eneigy functionals, which are certainly not universal. By focusing on the energy functional one has given up the connection to established quantum mechanics, which employs Hamiltonians and Hilbert spaces. One has then also abandoned the tradition of quantum chemistry of the development of hierarchies of approximations, which allows for step-wise systematic improvements of the description of electronic properties. 

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