Hamiltonian, total

A fundamental mathematical feature of chemical systems is that the relevant Hamiltonian (total energy) operator can be written in terms of one-electron ) and two-electron (Pop44) operators only. This implies that all relevant chemical information can he obtained from reduced density operators that condense and simplify the iV-electron... 

Physically, why does a temi like the Darling-Dennison couplmg arise We have said that the spectroscopic Hamiltonian is an abstract representation of the more concrete, physical Hamiltonian fomied by letting the nuclei in the molecule move with specified initial conditions of displacement and momentum on the PES, with a given total kinetic plus potential energy. This is the sense in which the spectroscopic Hamiltonian is an effective Hamiltonian, in the nomenclature used above. The concrete Hamiltonian that it mimics is expressed in temis of particle momenta and displacements, in the representation given by the nomial coordinates. Then, in general, it may contain temis proportional to all the powers of the products of the... 

The characteristic of the Darlhig-Deimison couplmg is that it exchanges two quanta between the synmietric and antisynmietric stretches. This means that the individual quantum mimbers n are no longer good quaiitum mimbers of the Hamiltonian containing However, the total iiumber of stretch quanta... 

We have seen that resonance couplings destroy quantum numbers as constants of the spectroscopic Hamiltonian. Widi both the Darling-Deimison stretch coupling and the Femii stretch-bend coupling in H2O, the individual quantum numbers and were destroyed, leaving the total polyad number n + +... 

We consider an isolated molecule in field-free space with Hamiltonian //. We let Pbe the total angular momentum operator of the molecule, that is... 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

Nomially the amplitude of the total incident field (or intensity of the incident light) is such that the light/matter coupling energies are sufficiently weak not to compete seriously with the dark matter Hamiltonian. As already noted, when this is tire case, tlie induced polarization, P is treated perturbatively in orders of the total electric field. Thus one writes... 

The total interaction energy of the nucleus may be expressed as a sum of the individual Hamiltonians given in equation B1.12.1, (listed in table B1.12.1) and are discussed in detail in several excellent books [1, 2, 3 and 4]. 

QMC teclmiques provide highly accurate calculations of many-electron systems. In variational QMC (VMC) [112, 113 and 114], the total energy of the many-electron system is calculated as the expectation value of the Hamiltonian. Parameters in a trial wavefiinction are optimized so as to find the lowest-energy state (modem... 

If tire mean Aq is positive tlien tire majority of tire residues are hydrophilic. A description of tire collapsed phase of tire chain requires introducing tliree- and and four-body interaction tenns. Thus, tire total Hamiltonian is... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

Thus, the total effective Hamiltonian (H) in the presence of a vector potential is now defined and it is for an type reactive system (9q = 0) given by... 

The total effective Hamiltonian H, in the presence of a vector potential for an A + B2 system is defined in Section II.B and the coupled first-order Hamilton equations of motion for all the coordinates are derived from the new effective Hamiltonian by the usual prescription [74], that is. 

Considering the semiclassical Hamiltonian from Eq. (28), one can expand the total wave function as. 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

The total elechon-nuclear Hamiltonian of a molecular sytem is defined as... 

If V is the total Coulombic potential between all the nuclei and electrons in the system, then, in the absence of any spin-dependent terms, the electronic Hamiltonian is given by... 

For a molecular system, we shall sepaiate the total Hamiltonian into three parts ... 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

The presence of two angular momenta has as a consequence that only their sum, representing the total angular momentum in the case considered, necessary commutes with the Hamiltonian of the system. Thus only the quantum number K, associated with the sum, N, of and Lj,... 

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