Approximation minimal coupling

The ab initio HF calculations reported below have been performed with the GAUSSIAN 76 [26] program package. The atomic basis sets applied are a minimal (STO-3G [26]) one, a split valence (6-31G [26]) one, a split-valence one plus a set of five d-functions on carbon (6-31G [26]), and one with an additional set of p-functions on hydrogen (6-31G [26]). The correlation energy has been computed using Mpller-Plesset many body perturbation theory of second order (MP2) [27], the linear approximation of Coupled Cluster Doubles theory (L-CCD)... 

In the Bom approximation retained here [3], an the minimal coupling framework used for the sake of simplicity leads to... 

This system in which the longitudinal and transverse motions of the emitter are separated to a good approximation, provides a convenient example for the consideration of the motional effects discussed above. As shown by Healy", the canonical transfoimiation from the minimal-coupling to the multipolar Hamiltonian has the same form for any convenient reference-point R (not necessarily the center-of-mass) relative to which the polarizations are defined. This arbitrariness, which amounts to a gauge... 

Fig. 7.7 Approximate nonadiabatic coupling vector Xgi (see the text for the definition) at t = 18.625 fs, at which the nonadiabatic coupling term reaches its maximum value in the collision of a water dimer anion and a water monomer. Although the vectors are time-dependent, their spatial orientations change only minimally during the periods of the vibration motions. (Reprinted with permission from T. Yonehara et al, Chem. Rev. 112, 499 (2012)).

Therefore, we will make a series of approximations to this approach. First, we will only use quantum mechanics for the description of the molecule and use classical electrodynamics for the electromagnetic fields. In this semi-classical approach the perturbing fields and nuclear moments are considered to be unaffected by the molecular environment, the so-called minimal coupling approximation. 

In the previous sections it was shown that in the minimal coupling approximation the vector potential enters the mechanical momentum of electron i... 

We consider a molecule in which the magnetic field arises from two primary sources, an external magnetic field induction and the permanent magnetic moments of nuclei possessing a spin. In the minimal coupling approximation, the mechanical momentum operator (O Eq. 11.23) is given as... 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

There are approximately 2700 compounds per primary screening mixture, and the readout is in essence multiplexed the ligands are individually ionized and identified in the mass spectrometer according to their exact mass positions. The readout, however, does not unambiguously identify compounds, as multiple compounds in a single mixture may have the same mass, i.e., a particular peak may correspond to as many as 31 compounds with closely related masses. The protein excess over individual compounds coupled with the rarity of potent ligands within a randomly assembled library minimizes competition between ligands for... 

In ab initio methods the HER approximation is used for build-up of initial estimate for and which have to be further improved by methods of configurational interaction in the complete active space (CAS) [39], or by Mpller-Plesset perturbation theory (MPn) of order n, or by the coupled clusters [40,41] methods. In fact, any reasonable result within the ab initio QC requires at least minimal involvement of electron correlation. All the technical tricks invented to go beyond the HFR calculation scheme in terms of different forms of the trial wave function or various perturbative procedures represent in fact attempts to estimate somehow the second term of Eq. (5) - the cumulant % of the two-particle density matrix. 

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