Molecular function time-dependent Schrodinger equation

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 ... 

We start from the time-dependent Schrodinger equation for the wave-function describing the evolution of the unperturbed molecular system,... 

The time evolution of molecular systems - i.e. systems containing particles of atomic dimensions ( electrons, nuclei etc) - can be adequately described within the framework of time-dependent quantum mechanics. All information about the system is contained in the time dependent wave function 0(r,r2,...,rN t) wherein the Ti are the position vectors (possibly containing also a spin component) for the i-th particle and t is the time. The wave function is a solution of the time-dependent Schrodinger equation... 

Selloni et al. [48] were the first to simulate adiabatic ground state quantum dynamics of a solvated electron. The system consisted of the electron, 32 K+ ions, and 31 Cl ions, with electron-ion interactions given by a pseudopotential. These simulations were unusual in that what has become the standard simulating annealing molecular dynamics scheme, described in the previous section, was not used. Rather, the wave function of the solvated electron was propagated forward in time with the time-dependent Schrodinger equation,... 

When the external electric field is time-dependent, there is no well-defined energy of the molecular system in accordance with Eq. (100), and the wave function response can thus not be retrieved from a variational condition on the energy as in the analytic derivative approach described above. Instead the response parameters have to be determined from the time-dependent Schrodinger equation, a procedure which was illustrated in Section 3 for the exact state case. In approximate state theories, however, our wave function space only partially spans the 7V-electron Hilbert space, and the response functions that correspond to an approximate state wave function will clearly be separate from those of the exact state wave function. This fact is disregarded in the sum-over-states approach, and, apart from the computational aspect of slowly converging SOS expressions, it is of little concern when highly accurate wave function models are used. But for less flexible wave function models, the correct response functions should be used in the calculation of nonlinear optical properties. 

To discriminate Car-Parrinello molecular dynamics more clearly, let us compare it with two other approaches. Ehren-fest molecular dynamics is based on the time-dependent Schrodinger equation for which the wave function stays minimized throughout the evolution of time such that the maximum time step is very limited be-... 

The free-energy profile is calculated by the FEP/US method (see section 16.3.3.3). However, at each step of the molecular dynamics simulation, the vibrational energy and the wave function of the transferred proton are determined from a three-dimensional Schrodinger equation and are included in the FEP/US procedure. In addition, dynamical effects due to transitions among proton vibrational states are calculated with a molecular dynamics with quantum transition (MDQT) procedure in which the proton wave function evolution is determined by a time-dependent Schrodinger equation. This procedure is combined with a reactive flux approach to calculate the transmission... 

An interesting application of the first case (i.e., an external oscillating field) is the study of the nonlinear properties of molecules in condensed matter. Once the approximate solutions of the corresponding time-dependent SchrOdinger equation are found, the frequency-dependent electric response functions (polarizability and hyperpolarizabilities tensors) of the molecular solute are easily calculated. 

Our task is to find approximate solutions to the time-independent Schrodinger equation (Eq. (2)) subject to the Pauli antisymmetry constraints of many-electron wave functions. Once such an approximate solution has been obtained, we may extract from it information about the electronic system and go on to compute different molecular properties related to experimental observations. Usually, we must explore a range of nuclear configurations in our calculations to determine critical points of the potential energy surface, or to include the effects of vibrational and rotational motions on the calculated properties. For properties related to time-dependent perturbations (e.g., all interactions with radiation), we must determine the time development of the... 

Even when confining the variation of the trial wavefunction to the LCAO-MO coefficients c U, the respective approximate solution of the Schrodinger equation is still quite complex and may be computationally very demanding. The major reason is that the third term of the electronic Hamiltonian, Hel (Equation 6.12), the electron-electron repulsion, depends on the coordinates of two electrons at a time, and thus cannot be broken down into a sum of one-electron functions. This contrasts with both the kinetic energy and the electron-nucleus attraction, each of which are functions of the coordinates of single electrons (and thus are written as sums of n one-electron terms). At the same time, orbitals are one-electron functions, and molecular orbitals can be more easily generated as eigenfunctions of an operator that can also be separated into one-electron terms. 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

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